2016年5月29日日曜日

160529(3)

整数列のLINKS の編集(28)

https://oeis.org/A006571
https://oeis.org/A030200
のLINKS を編集しました。

160529(2)

整数列のLINKS の編集(27)

https://oeis.org/A094343
https://oeis.org/A140447
のLINKS を編集しました。

160529

整数列のLINKS の編集(26)

https://oeis.org/A273221
https://oeis.org/A273222
https://oeis.org/A273223
https://oeis.org/A273224
のLINKS を編集しました。

2016年5月28日土曜日

160528(2)

Ruby


Bell number(2)

せきゅーんさんの記事(http://integers.hatenablog.com/entry/2016/05/28/025945)を見て、
Bell number n mod n を計算しようと思った。

オンライン整数列大辞典の
A166226(http://oeis.org/A166226/list)
と比較し、答え合わせしてみる。

def bell(n)
  return 1 if n == 0
  i = 1
  bell = [1]
  while i < n
    next_bell = [bell[-1]]
    # Bell triangle
    i.times{|j| next_bell[j + 1] = next_bell[j] + bell[j]}
    bell = next_bell
    i += 1
  end
  bell[-1]
end

def A166226(n)
  (1..n).map{|i| bell(i) % i}
end
ary = A166226(85)

# OEIS A166226のデータ
ary0 =
[0,0,2,3,2,5,2,4,6,5,2,1,2,12,5,3,2,13,2,12,15,5,
 2,9,3,18,10,3,2,27,2,12,4,5,0,1,2,24,28,27,2,23,2,
 8,5,5,2,33,24,20,49,39,2,5,27,28,34,5,2,57,2,36,6,
 51,47,19,2,52,15,25,2,49,2,42,22,71,59,19,2,44,23,
 5,2,65,84]
# 一致の確認
p ary == ary0

160528

整数列のLINKS の編集(25)

https://oeis.org/A035174
https://oeis.org/A064556
のLINKS を編集しました。

2016年5月27日金曜日

160527(2)

整数列のLINKS の編集(24)

https://oeis.org/A273465
のLINKS を編集しました。

160527

整数列のLINKS の編集(23)

https://oeis.org/A066481
https://oeis.org/A066482
のLINKS を編集しました。

2016年5月26日木曜日

160526(3)

整数列のLINKS の編集(22)

https://oeis.org/A272850
のLINKS を編集しました。

160526(2)

整数列のLINKS の編集(21)

https://oeis.org/A273372
https://oeis.org/A273373
https://oeis.org/A273374
https://oeis.org/A273375
のLINKS を編集しました。

160526

新たな整数列(5)

https://oeis.org/A272896
が追加されました。

2016年5月24日火曜日

160524(3)

Ruby


(number of divisors of n of form 4m + 1) - (number of divisors of form 4m + 3)

オンライン整数列大辞典の
A001826(http://oeis.org/A001826/list)、
A001842(http://oeis.org/A001842/list)、
A002654(http://oeis.org/A002654/list)
と比較し、答え合わせしてみる。

def r1(n)
  (1..n).select{|i| n % i == 0 && i % 4 == 1}.size
end

def r3(n)
  (1..n).select{|i| n % i == 0 && i % 4 == 3}.size
end

def A001826(n)
  (1..n).map{|i| r1(i)}
end

def A001842(n)
  # 0から始まる
  (0..n).map{|i| r3(i)}
end

def A002654(n)
  a, b = A001826(n), A001842(n)[1..-1]
  (0..n - 1).map{|i| a[i] - b[i]}
end

ary = A001826(105)
# OEIS A001826のデータ
ary0 =
[1,1,1,1,2,1,1,1,2,2,1,1,2,1,2,1,2,2,1,2,2,1,1,1,
 3,2,2,1,2,2,1,1,2,2,2,2,2,1,2,2,2,2,1,1,4,1,1,1,2,
 3,2,2,2,2,2,1,2,2,1,2,2,1,3,1,4,2,1,2,2,2,1,2,2,2,
 3,1,2,2,1,2,3,2,1,2,4,1,2,1,2,4,2,1,2,1,2,1,2,2,3,
 3,2,2,1,2,4]
# 一致の確認
p ary == ary0

ary = A001842(86)
# OEIS A001842のデータ
ary0 =
[0,0,0,1,0,0,1,1,0,1,0,1,1,0,1,2,0,0,1,1,0,2,1,1,
 1,0,0,2,1,0,2,1,0,2,0,2,1,0,1,2,0,0,2,1,1,2,1,1,1,
 1,0,2,0,0,2,2,1,2,0,1,2,0,1,3,0,0,2,1,0,2,2,1,1,0,
 0,3,1,2,2,1,0,2,0,1,2,0,1]
# 一致の確認
p ary == ary0

ary = A002654(105)
# OEIS A002654のデータ
ary0 =
[1,1,0,1,2,0,0,1,1,2,0,0,2,0,0,1,2,1,0,2,0,0,0,0,
 3,2,0,0,2,0,0,1,0,2,0,1,2,0,0,2,2,0,0,0,2,0,0,0,1,
 3,0,2,2,0,0,0,0,2,0,0,2,0,0,1,4,0,0,2,0,0,0,1,2,2,
 0,0,0,0,0,2,1,2,0,0,4,0,0,0,2,2,0,0,0,0,0,0,2,1,0,
 3,2,0,0,2,0]
# 一致の確認
p ary == ary0

出力結果
true
true
true

160524(2)

新たな整数列(4)

https://oeis.org/A272898
が追加されました。

160524

整数列のLINKS の編集(20)

https://oeis.org/A140445
https://oeis.org/A140446
https://oeis.org/A272816
https://oeis.org/A153419
のLINKS を編集しました。

2016年5月22日日曜日

160522(3)

Ruby


Π(1 - q^n)(2)

オイラーの五角数定理を使うと高速に求まる。

オンライン整数列大辞典の
A010815(http://oeis.org/A010815/list)
と比較し、答え合わせしてみる。

def p0(n)
  (3 * n * n - n) / 2
end

def p1(n)
  (3 * n * n + n) / 2
end

def A010815(n)
  ary = Array.new(n + 1, 0)
  ary[0] = 1
  i = 1
  j = p0(i)
  while j <= n
    ary[j] = (-1) ** i
    i += 1
    j = p0(i)
  end
  i = 1
  j = p1(i)
  while j <= n
    ary[j] = (-1) ** i
    i += 1
    j = p1(i)
  end
  ary
end
ary = A010815(92)

# OEIS A010815のデータ
ary0 =
[1,-1,-1,0,0,1,0,1,0,0,0,0,-1,0,0,-1,0,0,0,0,0,0,
 1,0,0,0,1,0,0,0,0,0,0,0,0,-1,0,0,0,0,-1,0,0,0,0,0,
 0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,
 -1,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1]
# 一致の確認
p ary == ary0

160522(2)

新たな整数列(3)

https://oeis.org/A272697
https://oeis.org/A272698
が追加されました。

160522

整数列のLINKS の編集(19)

https://oeis.org/A030205
https://oeis.org/A159817
のLINKS を編集しました。

2016年5月21日土曜日

160521

整数列のLINKS の編集(18)

https://oeis.org/A049488
https://oeis.org/A272815
のLINKS を編集しました。

2016年5月20日金曜日

160520

Ruby


Π(1 - q^n)(1)

オンライン整数列大辞典の
A010815(http://oeis.org/A010815/list)
と比較し、答え合わせしてみる。

# m次以下を取り出す
def mul(f_ary, b_ary, m)
  s1, s2 = f_ary.size, b_ary.size
  ary = Array.new(s1 + s2 - 1, 0)
  (0..s1 - 1).each{|i|
    (0..s2 - 1).each{|j|
      ary[i + j] += f_ary[i] * b_ary[j]
    }
  }
  ary[0..m]
end

def A010815(n)
  ary = [1]
  (1..n).each{|i|
    b_ary = Array.new(i + 1, 0)
    b_ary[0], b_ary[-1] = 1, -1
    ary = mul(ary, b_ary, n)
  }
  ary
end
ary = A010815(92)

# OEIS A010815のデータ
ary0 =
[1,-1,-1,0,0,1,0,1,0,0,0,0,-1,0,0,-1,0,0,0,0,0,0,
 1,0,0,0,1,0,0,0,0,0,0,0,0,-1,0,0,0,0,-1,0,0,0,0,0,
 0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,
 -1,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1]
# 一致の確認
p ary == ary0

2016年5月19日木曜日

160519(2)

整数列のLINKS の編集(17)

https://oeis.org/A272569
のLINKS を編集しました。

160519

Ruby


Ménage Number

オンライン整数列大辞典の
A000179(http://oeis.org/A000179/list)
と比較し、答え合わせしてみる。

def a(n)
  return 1 if n <= 1
  (1..n).inject(:*)
end

def c(n, r)
  r = n - r if r > n - r
  return 0 if r < 0
  return 1 if r == 0
  return (n - r + 1..n).inject(:*) / (1..r).inject(:*)
end

def f(n)
  (0..n).inject(0){|s, i| s += (-1) ** i * a(n - i) * c(2 * n - i, i) * 2 * n / (2 * n - i)}
end

def A000179(n)
  ary = [1, 0]
  return ary[0..n] if n <= 1
  ary + (2..n).map{|i| f(i)}
end
ary = A000179(25)

# OEIS A000179のデータ
ary0 =
[1,0,0,1,2,13,80,579,4738,43387,439792,4890741,
 59216642,775596313,10927434464,164806435783,
 2649391469058,45226435601207,817056406224416,
 15574618910994665,312400218671253762,
 6577618644576902053,145051250421230224304,
 3343382818203784146955,80399425364623070680706,
 2013619745874493923699123]
# 一致の確認
p ary == ary0

2016年5月18日水曜日

160518(2)

整数列のLINKS の編集(16)

https://oeis.org/A072960
のLINKS を編集しました。

160518

Ruby


Goldbach's conjecture(4)

x^(2 + 2) + (x^5 + x^13 + x^17 + … )(x^3 + x^7 + x^11 + x^19 + … )
= x^4 + x^8 + x^12 + 2x^16 + 2x^20 + 3x^24 + …

x^(4 * n) の係数が正
ということが成り立つとして、
その係数を列挙してみる。

オンライン整数列大辞典の
A069360(http://oeis.org/A069360/list)
と比較し、答え合わせしてみる。

require 'prime'

# m次以下を取り出す
def mul(f_ary, b_ary, m)
  s1, s2 = f_ary.size, b_ary.size
  ary = Array.new(s1 + s2 - 1, 0)
  s10 = [s1 - 1, m].min
  (0..s10).each{|i|
    s20 = [s2 - 1, m - i].min
    (0..s20).each{|j|
      ary[i + j] += f_ary[i] * b_ary[j]
    }
  }
  ary
end

def A069360(n)
  m = 4 * n
  ary1 = Array.new(m + 1, 0)
  Prime.each(m).select{|i| i % 4 == 1}.each{|i| ary1[i] = 1}
  ary3 = Array.new(m + 1, 0)
  Prime.each(m).select{|i| i % 4 == 3}.each{|i| ary3[i] = 1}
  a = mul(ary1, ary3, m)[0..m]
  a.delete(0)
  # (2 + 2) / 2 = 2 * 1の分
  [1] + a
end
ary = A069360(80)

# OEIS A069360のデータ
ary0 =
[1,1,1,2,2,3,2,2,4,3,3,5,3,3,6,5,2,6,5,4,8,4,4,7,
 6,5,8,7,6,12,5,3,9,5,7,11,5,4,11,8,5,13,6,7,14,8,
 5,11,9,8,14,7,6,13,9,7,12,7,9,18,9,6,16,8,10,16,9,
 7,16,14,8,17,8,8,21,10,8,17,10,11]
# 一致の確認
p ary == ary0

2016年5月17日火曜日

160517(2)

Ruby


Goldbach's conjecture(3)

n = 10 ** i のとき、ペアの数を数えてみる。

require'prime'

def f(n)
  Prime.each(n / 2).count{|i| (n - i).prime?}
end

def A065577(n)
  (1..n).map{|i| f(10 ** i)}
end

p A065577(7)

出力結果
[2, 6, 28, 127, 810, 5402, 38807]

上は短いコードだが大変遅い。
ちなみに、次のコードはかなり速い。

def p_table(n)
  ary = Array.new(n + 1, true)
  ary[0], ary[1] = false, false
  i = 2
  while i * i <= n
    if ary[i]
      j = i + i
      while j <= n
        ary[j] = false
        j += i
      end
    end
    i += 1
  end
  ary
end

def f(n)
  a = p_table(n)
  (1..n / 2).count{|i| a[i] && a[n - i] == true}
end

def A065577(n)
  (1..n).map{|i| f(10 ** i)}
end

p A065577(8)

出力結果
[2, 6, 28, 127, 810, 5402, 38807, 291400]

160517

Ruby


Goldbach's conjecture(2)

ペアの数を数えてみた。、
オンライン整数列大辞典の
A061358(http://oeis.org/A061358/list)
と比較し、答え合わせしてみる。

require'prime'

def A061358(n)
  (0..n).map{|i| Prime.each(i / 2).count{|j| (i - j).prime?}}
end
ary = A061358(105)

# OEIS A061358のデータ
ary0 =
[0,0,0,0,1,1,1,1,1,1,2,0,1,1,2,1,2,0,2,1,2,1,3,0,
 3,1,3,0,2,0,3,1,2,1,4,0,4,0,2,1,3,0,4,1,3,1,4,0,5,
 1,4,0,3,0,5,1,3,0,4,0,6,1,3,1,5,0,6,0,2,1,5,0,6,1,
 5,1,5,0,7,0,4,1,5,0,8,1,5,0,4,0,9,1,4,0,5,0,7,0,3,
 1,6,0,8,1,5,1]
# 一致の確認
p ary == ary0

2016年5月16日月曜日

160516(4)

Ruby


Number of ordered ways of writing n as sum of three primes

(Σx^p)^3 の係数について、
オンライン整数列大辞典の
A098238(http://oeis.org/A098238/list)
と比較し、答え合わせしてみる。

require 'prime'

# m次以下を取り出す
def mul(f_ary, b_ary, m)
  s1, s2 = f_ary.size, b_ary.size
  ary = Array.new(s1 + s2 - 1, 0)
  s10 = [s1 - 1, m].min
  (0..s10).each{|i|
    s20 = [s2 - 1, m - i].min
    (0..s20).each{|j|
      ary[i + j] += f_ary[i] * b_ary[j]
    }
  }
  ary
end

def A098238(n)
  ary = Array.new(n + 1, 0)
  Prime.each(n).each{|i| ary[i] = 1}
  mul(mul(ary, ary, n), ary, n)[0..n]
end
ary = A098238(74)

# OEIS A098238のデータ
ary0 =
[0,0,0,0,0,0,1,3,3,4,6,6,9,6,6,10,9,12,12,12,12,
 19,12,21,15,21,18,30,15,30,12,30,18,37,12,39,21,
 42,24,46,9,51,18,48,24,54,18,66,21,60,30,67,24,81,
 18,75,30,79,18,87,21,87,36,93,15,105,30,105,36,97,
 12,120,30,114,36]
# 一致の確認
p ary == ary0

160516(3)

Ruby


Goldbach's conjecture(1)

この予想を組合せ論っぽく言い換えると次のようになる。

p が素数全体を動くとき、
(Σx^p)^2 の2 よりも大きな偶数次の係数は必ず1 以上。

(Σx^p)^2 の係数について、
オンライン整数列大辞典の
A073610(http://oeis.org/A073610/list)
と比較し、答え合わせしてみる。

require 'prime'

# m次以下を取り出す
def mul(f_ary, b_ary, m)
  s1, s2 = f_ary.size, b_ary.size
  ary = Array.new(s1 + s2 - 1, 0)
  s10 = [s1 - 1, m].min
  (0..s10).each{|i|
    s20 = [s2 - 1, m - i].min
    (0..s20).each{|j|
      ary[i + j] += f_ary[i] * b_ary[j]
    }
  }
  ary
end

def A073610(n)
  ary = Array.new(n + 1, 0)
  Prime.each(n).each{|i| ary[i] = 1}
  mul(ary, ary, n)[1..n]
end
ary = A073610(98)

# OEIS A073610のデータ
ary0 =
[0,0,0,1,2,1,2,2,2,3,0,2,2,3,2,4,0,4,2,4,2,5,0,6,
 2,5,0,4,0,6,2,4,2,7,0,8,0,3,2,6,0,8,2,6,2,7,0,10,
 2,8,0,6,0,10,2,6,0,7,0,12,2,5,2,10,0,12,0,4,2,10,
 0,12,2,9,2,10,0,14,0,8,2,9,0,16,2,9,0,8,0,18,2,8,
 0,9,0,14,0,6]
# 一致の確認
p ary == ary0

160516(2)

Ruby


獣の数字(2)

2 + 3 + 5 + 7 + 11 + 13 + 17 + 19 = 77
2^2 + 3^2 + 5^2 + 7^2 + 11^2 + 13^2 + 17^2 = 666
が成り立つ。
他にp 以下の素数のn 乗の和がゾロ目になる数を探してみたが見つからなかった。

require 'prime'

(1..100).each{|n|
  s = 0
  Prime.each(10 ** 4).each{|i|
    s += i ** n
    p [n, i, s] if s.to_s.split('').to_a.uniq.size == 1
  }
}

出力結果
[1, 2, 2]
[1, 3, 5]
[1, 19, 77]
[2, 2, 4]
[2, 17, 666]
[3, 2, 8]

160516

Ruby


獣の数字(1)

2^2 + 3^2 + 5^2 + 7^2 + 11^2 + 13^2 + 17^2 = 666
が成り立つ。
他にp 以下の素数の二乗の和がゾロ目になる数を探してみたが見つからなかった。

require 'prime'

s = 0
Prime.each(10 ** 8).each{|i|
  s += i * i
  p [i, s] if s.to_s.split('').to_a.uniq.size == 1
}

出力結果
[2, 4]
[17, 666]

2016年5月15日日曜日

160515(4)

Ruby


Two refinements of Lagrange's four-square theorem

せきゅーんさんのブログに紹介があった
以下のプレプリントに興味を持った。
https://arxiv.org/pdf/1605.03074v2.pdf

任意の自然数n に対し、
ある整数x, y, z, w が存在し、
n = x^2 + y^2 + z^2 + w^2,
x + y + z + w は平方数
とできることを証明している。

例えば、
160515 = 37249 + 39601 + 40401 + 43264 = 193^2 + (-199)^2 + (-201)^2 + 208^2,
193 - 199 - 201 + 208 = 1 = 1^2
とできる。

各n に対し、このようなx, y, z, w を見つけてみた。

def square?(n)
  return false if n < 0
  Math.sqrt(n).to_i ** 2 == n
end

def f(ary)
  x, y, z, w = ary
  # マイナス符号のつけかたを考えるのは最小限にとどめる。
  [[ x,  y,  z,  w],
   [ x,  y,  z, -w],
   [ x,  y, -z,  w],
   [ x,  y, -z, -w],
   [ x, -y,  z,  w],
   [ x, -y,  z, -w],
   [ x, -y, -z,  w],
   [ x, -y, -z, -w],
   [-x,  y,  z,  w],
   [-x,  y,  z, -w],
   [-x,  y, -z,  w],
   [-x,  y, -z, -w],
   [-x, -y,  z,  w],
   [-x, -y,  z, -w],
   [-x, -y, -z,  w],
   [-x, -y, -z, -w]].each{|a|
    return [1, a] if square?(a.inject(:+))
  }
  [0, ary]
end

def g(n)
  ary = []
  (0..Math.sqrt(n).to_i).each{|x|
    (x..Math.sqrt(n - x * x).to_i).each{|y|
      (y..Math.sqrt(n - x * x - y * y).to_i).each{|z|
        w2 = n - x * x - y * y - z * z
        if w2 >= z * z && square?(w2)
          a = f([x, y, z, Math.sqrt(w2).to_i])
          ary << a[1] if a[0] == 1
        end
      }
    }
  }
  ary
end

(1..200).each{|i| p [i, g(i)]}

出力結果
[1, [[0, 0, 0, 1]]]
[2, [[0, 0, 1, -1]]]
[3, [[0, 1, 1, -1]]]
[4, [[1, 1, 1, 1]]]
[5, [[0, 0, -1, 2]]]
[6, [[0, 1, 1, 2]]]
[7, [[1, 1, 1, -2]]]
[8, [[0, 0, 2, 2]]]
[9, [[0, 1, 2, -2]]]
[10, [[0, 0, 1, 3], [1, -1, 2, 2]]]
[11, [[0, -1, -1, 3]]]
[12, [[1, 1, 1, -3]]]
[13, [[0, 0, -2, 3], [-1, 2, 2, -2]]]
[14, [[0, 1, 2, -3]]]
[15, [[1, 1, 2, -3]]]
[16, [[0, 0, 0, 4], [2, 2, 2, -2]]]
[17, [[0, 2, 2, -3]]]
[18, [[0, 0, 3, -3], [0, 1, -1, 4], [1, 2, -2, 3]]]
[19, [[0, 1, 3, -3], [-1, -1, -1, 4]]]
[20, [[1, -1, 3, -3]]]
[21, [[0, -1, -2, 4], [2, 2, 2, 3]]]
[22, [[0, -2, 3, 3], [1, 1, 2, -4]]]
[23, [[1, 2, 3, 3]]]
[24, [[0, 2, 2, -4]]]
[25, [[0, 0, -3, 4], [1, 2, 2, 4]]]
[26, [[0, 0, -1, 5], [0, 1, 3, -4], [2, 2, 3, -3]]]
[27, [[0, 3, 3, 3], [1, 1, 3, 4]]]
[28, [[1, -1, -1, 5], [1, 3, 3, -3]]]
[29, [[0, 2, 3, 4]]]
[30, [[0, 1, -2, 5], [1, 2, -3, 4]]]
[31, [[1, 1, 2, 5], [-2, 3, 3, -3]]]
[32, [[0, 0, 4, -4]]]
[33, [[0, 1, 4, 4], [0, 2, 2, 5], [2, -2, -3, 4]]]
[34, [[0, 3, -3, 4], [1, -1, 4, -4], [1, 2, 2, -5]]]
[35, [[0, 1, 3, 5], [-1, 3, 3, 4]]]
[36, [[1, 1, 3, -5], [3, 3, -3, -3]]]
[37, [[-1, 2, 4, 4], [2, 2, 2, -5]]]
[38, [[0, -1, -1, 6], [0, 2, 3, -5], [2, 3, 3, -4]]]
[39, [[1, 1, 1, 6], [1, 2, 3, -5]]]
[40, [[0, 0, -2, 6], [2, 2, 4, -4]]]
[41, [[0, 0, 4, 5], [0, 1, 2, 6]]]
[42, [[0, 1, 4, -5], [1, -1, -2, 6], [1, 3, 4, -4], [-2, -2, 3, 5]]]
[43, [[0, 3, 3, -5], [1, 1, 4, -5], [3, -3, -3, 4]]]
[44, [[1, -3, -3, 5]]]
[45, [[0, 0, 3, 6], [0, 2, 4, -5], [-1, 2, 2, 6], [-2, 3, 4, 4]]]
[46, [[0, 1, -3, 6], [1, 2, -4, 5]]]
[47, [[1, -1, 3, 6], [2, -3, -3, 5]]]
[48, [[0, 4, 4, -4], [2, 2, 2, -6]]]
[49, [[0, -2, -3, 6], [2, -2, 4, 5]]]
[50, [[0, 0, 5, -5], [0, 3, -4, 5], [1, 2, 3, -6], [3, -3, 4, -4]]]
[51, [[0, 1, 1, 7], [0, 1, 5, -5], [-1, 3, 4, -5]]]
[52, [[-1, -1, -1, 7], [1, -1, 5, -5], [3, 3, 3, -5]]]
[53, [[0, 0, 2, 7], [0, -1, 4, 6], [2, 2, 3, -6]]]
[54, [[0, -1, -2, 7], [0, 3, 3, -6], [1, 1, 4, -6], [2, 3, 4, -5]]]
[55, [[1, -1, 2, 7], [1, -2, 5, 5], [1, 3, 3, -6]]]
[56, [[0, 2, 4, -6]]]
[57, [[1, 2, 4, -6], [-3, 4, 4, 4]]]
[58, [[0, 0, -3, 7], [1, -2, -2, 7], [1, 4, 4, -5], [2, 2, 5, -5], [-2, 3, -3, 6]]]
[59, [[0, -1, 3, 7], [3, -3, 4, 5]]]
[60, [[1, -1, -3, 7], [1, 3, 5, -5]]]
[61, [[0, 0, -5, 6], [0, 3, 4, -6], [2, 2, -2, 7], [-2, 4, 4, -5]]]
[62, [[0, 1, 5, -6], [1, -3, -4, 6]]]
[63, [[1, 1, 5, -6], [1, -2, 3, 7], [2, -3, 5, 5], [3, 3, -3, 6]]]
[64, [[4, 4, 4, 4]]]
[65, [[0, 0, 1, 8], [0, 2, 5, -6], [2, -3, 4, 6]]]
[66, [[0, 1, -4, 7], [0, 4, 5, -5], [1, 2, -5, 6], [2, 2, 3, -7], [3, 4, 4, 5]]]
[67, [[0, -3, -3, 7], [1, 1, -1, 8], [-1, -1, 4, 7]]]
[68, [[1, 3, 3, -7], [3, 3, 5, 5]]]
[69, [[0, -1, 2, 8], [0, -2, 4, 7], [-1, 4, 4, -6], [2, -2, -5, 6]]]
[70, [[0, 3, -5, 6], [-1, -1, -2, 8], [1, 2, 4, -7], [2, 4, 5, 5], [3, 3, 4, 6]]]
[71, [[1, -3, 5, 6], [2, 3, 3, -7]]]
[72, [[0, 0, 6, -6], [0, -2, -2, 8], [2, 4, 4, 6]]]
[73, [[0, 1, 6, -6], [1, 2, -2, 8], [2, 2, 4, -7], [4, 4, -4, 5]]]
[74, [[0, -1, -3, 8], [0, 3, 4, -7], [1, -1, 6, -6], [2, 3, 5, 6]]]
[75, [[0, -1, -5, 7], [-1, -1, 3, 8], [1, 3, 4, -7], [3, -4, 5, 5]]]
[76, [[1, 1, 5, -7], [1, 5, 5, 5], [3, 3, 3, 7]]]
[77, [[0, -2, 3, 8], [-1, 2, 6, -6], [3, 4, -4, 6]]]
[78, [[0, 2, 5, -7], [1, -2, -3, 8], [1, 4, 5, 6], [2, 3, 4, 7]]]
[79, [[1, 2, 5, -7], [3, -3, -5, 6]]]
[80, [[0, 0, -4, 8], [2, 2, 6, 6]]]
[81, [[0, 0, 0, 9], [0, -3, 6, 6], [0, 4, 4, -7], [2, 2, -3, 8], [2, -4, 5, 6]]]
[82, [[1, -1, -4, 8], [1, 3, 6, 6], [1, 4, 4, 7], [2, 2, 5, 7], [4, -4, 5, -5]]]
[83, [[0, 1, -1, 9], [0, 3, 5, -7], [1, 3, -3, 8], [3, 3, -4, 7]]]
[84, [[1, 3, 5, 7]]]
[85, [[0, 0, -6, 7], [-1, -2, 4, 8], [-2, 3, 6, -6], [2, 4, -4, 7]]]
[86, [[0, 1, 6, -7], [0, 5, 5, 6], [2, 3, 3, 8], [3, -4, -5, 6]]]
[87, [[1, 1, -2, 9], [1, 1, 6, -7], [2, -3, -5, 7]]]
[88, [[0, 4, 6, 6], [2, 2, 4, 8]]]
[89, [[0, 2, -2, 9], [0, 2, 6, -7], [0, -3, 4, 8], [1, -4, 6, 6]]]
[90, [[0, 1, -5, 8], [0, 4, 5, 7], [-1, -2, -2, 9], [1, 2, 6, 7], [1, 3, 4, 8], [-2, 5, -5, 6], [3, -3, 6, -6], [3, 4, 4, -7]]]
[91, [[-1, -1, -5, 8], [1, -4, 5, 7], [3, 3, 3, -8], [4, 5, 5, -5]]]
[92, [[-1, -1, -3, 9], [3, 3, 5, -7]]]
[93, [[0, -2, -5, 8], [2, -2, -6, 7], [2, 3, 4, -8], [4, 4, -5, 6]]]
[94, [[0, -2, -3, 9], [0, 3, 6, 7], [1, 2, 5, 8], [2, 4, 5, -7]]]
[95, [[1, 2, -3, 9], [-1, 3, 6, -7], [3, 5, -5, 6]]]
[96, [[0, 4, 4, 8]]]
[97, [[1, 4, 4, -8], [2, 2, 5, -8], [-3, 4, 6, -6]]]
[98, [[0, 0, 7, -7], [0, -1, -4, 9], [0, 3, 5, 8], [-1, 5, 6, 6], [2, 2, 3, 9], [2, 3, 6, -7], [3, -3, -4, 8]]]
[99, [[0, 1, 7, -7], [0, 3, -3, 9], [1, 3, 5, -8], [3, 4, -5, 7]]]
[100, [[1, 1, 7, 7], [1, 3, 3, 9], [1, 5, 5, -7], [5, 5, -5, -5]]]
[101, [[0, 0, -1, 10], [0, -1, -6, 8], [0, -4, 6, 7], [2, -5, 6, 6]]]
[102, [[0, 2, 7, 7], [1, 1, 6, 8], [1, 2, 4, 9], [1, 4, 6, -7], [-2, 3, -5, 8], [4, -5, -5, 6]]]
[103, [[1, -1, -1, 10], [-1, 2, 7, -7], [-2, -3, -3, 9], [2, 5, -5, 7], [3, -3, -6, 7]]]
[104, [[0, 2, 6, 8], [4, -4, 6, -6]]]
[105, [[0, 1, -2, 10], [0, 4, 5, -8], [1, 2, 6, -8], [2, 2, -4, 9], [-2, 4, 6, -7]]]
[106, [[0, 0, -5, 9], [0, 3, 4, 9], [1, -4, -5, 8], [2, 2, 7, -7], [-3, -5, 6, 6], [-4, -4, 5, 7]]]
[107, [[1, 3, -4, 9], [3, 3, -5, 8]]]
[108, [[1, 1, 5, 9], [1, 3, 7, -7], [3, 3, 3, -9], [3, -5, -5, 7]]]
[109, [[0, 3, 6, -8], [-1, 2, -2, 10], [2, 4, -5, 8]]]
[110, [[0, 2, 5, 9], [0, 5, 6, -7], [1, -3, -6, 8], [2, 3, 4, -9], [3, -4, -6, 7]]]
[111, [[1, 1, -3, 10], [-1, -2, -5, 9], [1, -5, 6, 7], [-2, 3, 7, -7], [5, 5, 5, -6]]]
[112, [[2, 2, 2, 10], [-2, 6, 6, 6], [4, 4, 4, -8]]]
[113, [[0, 0, -7, 8], [0, 2, -3, 10], [0, 4, -4, 9], [2, -3, -6, 8], [4, 5, 6, -6]]]
[114, [[0, 1, 7, 8], [0, 4, 7, -7], [1, 2, 3, 10], [1, 4, 4, -9], [2, 2, 5, -9], [-2, 5, 6, 7], [3, 4, 5, -8]]]
[115, [[0, -3, -5, 9], [1, 1, 7, -8], [-1, -4, 7, 7], [1, 5, -5, 8], [3, 3, 4, -9], [4, -5, -5, 7]]]
[116, [[1, 3, 5, -9], [3, -3, 7, -7]]]
[117, [[0, 2, 7, -8], [-1, 4, 6, -8], [-2, -2, 3, 10], [2, 4, 4, -9], [3, 6, 6, -6], [4, 4, -6, 7]]]
[118, [[0, 1, 6, 9], [0, 3, 3, 10], [1, 1, 4, 10], [1, 2, -7, 8], [-2, 4, 7, 7], [2, 5, 5, -8], [3, 3, 6, -8]]]
[119, [[-1, -1, -6, 9], [-1, 3, -3, 10], [2, 3, 5, -9], [3, 5, -6, 7]]]
[120, [[0, 2, 4, 10], [2, 4, 6, -8]]]
[121, [[0, -2, -6, 9], [1, 2, -4, 10], [2, -2, -7, 8]]]
[122, [[0, 3, -7, 8], [0, 4, 5, -9], [1, 2, 6, -9], [-1, 6, 6, -7], [3, -4, -4, 9], [5, -5, 6, -6]]]
[123, [[0, -1, -1, 11], [0, -5, 7, 7], [-1, 3, 7, -8], [1, 4, 5, -9], [-3, 4, 7, -7], [3, -5, -5, 8]]]
[124, [[-1, 5, 7, -7], [3, -3, -5, 9]]]
[125, [[0, 0, -2, 11], [0, 3, -4, 10], [0, -5, 6, 8], [2, 2, 6, -9], [2, 6, -6, 7], [3, 4, -6, 8]]]
[126, [[0, 1, 5, 10], [0, 3, 6, -9], [1, -3, -4, 10], [1, 5, 6, -8], [2, 3, 7, -8], [-2, 4, 5, 9], [4, -5, -6, 7]]]
[127, [[1, -1, -2, 11], [1, 3, 6, -9], [-3, -3, -3, 10]]]
[128, [[0, 0, 8, 8]]]
[129, [[0, 1, 8, -8], [-2, -3, 4, 10], [2, 5, -6, 8]]]
[130, [[0, 0, 7, 9], [1, -1, 8, 8], [1, 2, 2, 11], [1, -2, -5, 10], [1, 4, 7, -8], [-2, 3, 6, 9], [-3, 6, 6, 7], [4, -4, 7, -7], [-4, 5, -5, 8]]]
[131, [[0, 1, -3, 11], [0, -1, -7, 9], [0, 5, 5, -9], [3, -3, -7, 8]]]
[132, [[1, 1, 3, 11], [1, 1, 7, -9], [1, -5, -5, 9], [-3, 5, 7, 7]]]
[133, [[0, 4, 6, -9], [-1, 2, 8, -8], [-1, 4, -4, 10], [2, -2, -2, 11], [2, 2, -5, 10], [-2, 4, 7, -8], [-5, 6, 6, -6]]]
[134, [[0, 2, 3, 11], [0, 2, 7, -9], [1, -4, -6, 9], [3, 3, 4, -10], [3, -5, -6, 8]]]
[135, [[-1, 2, -3, 11], [1, 2, 7, -9], [1, 3, -5, 10], [1, -6, 7, 7], [2, -5, -5, 9], [3, 3, -6, 9], [5, 5, 6, -7]]]
[136, [[0, 0, 6, 10], [0, 6, 6, -8], [2, 2, 8, -8], [2, 4, 4, -10]]]
[137, [[1, 6, -6, 8], [2, 4, -6, 9], [4, 6, 6, -7]]]
[138, [[0, 1, 4, 11], [0, 5, 7, -8], [1, -1, 6, 10], [1, 3, 8, -8], [-2, -2, -3, 11], [2, -2, 7, 9], [2, 3, 5, -10], [-2, 6, 7, -7], [3, -4, -7, 8], [4, 4, 5, -9]]]
[139, [[0, 3, 7, -9], [1, 1, -4, 11], [-1, -5, 7, 8], [4, 5, 7, -7]]]
[140, [[-1, 3, 3, 11], [1, -3, -7, 9], [3, 5, 5, -9]]]
[141, [[0, 2, -4, 11], [0, 4, -5, 10], [-1, -2, -6, 10], [-2, 3, 8, -8], [3, 4, 4, -10], [4, -5, -6, 8]]]
[142, [[-1, 2, 4, 11], [1, 4, 5, -10], [-2, 5, -7, 8], [3, 4, 6, -9]]]
[143, [[1, 5, -6, 9], [-2, 3, -3, 11], [2, -3, -7, 9], [3, 3, 5, -10], [3, 6, 7, -7]]]
[144, [[0, 4, 8, -8], [2, 2, 6, -10], [6, 6, -6, -6]]]
[145, [[0, 0, -8, 9], [0, -3, -6, 10], [2, 4, 5, -10], [-3, 6, 6, -8], [4, 4, -7, 8]]]
[146, [[0, 0, 5, 11], [0, 1, 8, -9], [0, -3, -4, 11], [1, 3, 6, -10], [2, 5, 6, -9], [3, -3, 8, 8], [5, -6, -6, 7]]]
[147, [[-1, -1, -1, 12], [1, 1, 8, -9], [-1, 3, -4, 11], [-1, 4, 7, -9], [3, 5, -7, 8]]]
[148, [[1, -1, 5, 11], [3, 3, 7, -9], [5, -5, 7, -7]]]
[149, [[0, -1, -2, 12], [0, 2, 8, -9], [0, -6, 7, 8], [2, 3, 6, -10]]]
[150, [[0, 1, -7, 10], [0, -2, -5, 11], [0, 5, 5, -10], [1, 1, 2, 12], [1, 2, -8, 9], [-1, 6, 7, -8], [-2, 3, 4, 11], [2, 4, 7, -9], [3, -4, -5, 10], [-4, 6, 7, 7], [-5, -5, 6, 8]]]
[151, [[-1, -1, -7, 10], [1, 2, -5, 11], [1, 5, 5, -10], [2, 7, 7, -7], [3, -5, -6, 9]]]
[152, [[0, 2, 2, 12], [0, 4, 6, -10], [4, -6, -6, 8]]]
[153, [[0, 0, -3, 12], [0, -2, -7, 10], [0, 6, -6, 9], [1, -2, -2, 12], [1, 4, 6, -10], [2, -2, -8, 9], [2, 6, -7, 8], [-3, 4, 8, -8]]]
[154, [[0, 1, 3, 12], [0, 3, -8, 9], [1, 2, 7, -10], [1, -4, -4, 11], [-1, 5, 8, -8], [1, 6, 6, -9], [2, -2, 5, 11], [3, -3, 6, 10], [4, -5, -7, 8]]]
[155, [[0, 3, -5, 11], [1, -1, -3, 12], [-1, 3, 8, -9], [-3, -3, 4, 11], [3, 4, -7, 9]]]
[156, [[1, -3, -5, 11], [1, 5, 7, -9], [-3, 7, 7, -7], [5, -5, -5, 9]]]
[157, [[2, 2, 7, -10], [-2, 4, -4, 11], [-2, -5, 8, 8], [-2, 6, 6, -9], [6, 6, 6, 7]]]
[158, [[0, -1, 6, 11], [0, 3, 7, -10], [-1, 2, 3, 12], [2, 3, 8, -9], [-3, 6, -7, 8], [-4, 5, 6, 9]]]
[159, [[1, 3, 7, -10], [-2, -3, -5, 11], [2, 5, -7, 9], [5, 6, 7, 7]]]
[160, [[0, 0, 4, 12], [4, -4, 8, 8]]]
[161, [[0, 1, -4, 12], [0, 5, 6, -10], [2, -2, -3, 12], [-3, -4, 6, 10], [5, 6, 6, 8]]]
[162, [[0, 0, 9, -9], [1, -1, 4, 12], [1, -2, 6, 11], [1, 4, 8, -9], [1, -5, -6, 10], [2, -3, 7, 10], [3, 4, 4, -11], [3, -6, -6, 9], [4, -4, 7, 9]]]
[163, [[0, 1, 9, -9], [-1, 4, -5, 11], [1, 7, -7, 8], [3, -3, -8, 9], [4, 7, 7, 7], [5, 5, 7, 8]]]
[164, [[1, -1, 9, -9], [3, 3, 5, -11], [3, -5, -7, 9]]]
[165, [[0, -1, -8, 10], [0, 4, 7, -10], [-1, 2, -4, 12], [-1, -6, 8, 8], [2, 2, -6, 11], [-2, 4, 8, -9], [2, -5, -6, 10], [4, 6, 7, 8]]]
[166, [[0, -2, 9, 9], [0, 6, 7, -9], [1, 1, 8, -10], [1, -4, -7, 10], [-2, 3, 3, 12], [2, 4, 5, -11], [-2, 7, 7, -8], [4, 5, 5, -10]]]
[167, [[-1, 2, 9, -9], [1, 3, -6, 11], [1, 6, -7, 9], [3, 3, -7, 10], [5, 5, 6, 9]]]
[168, [[0, 2, 8, -10], [2, -2, 4, 12], [-2, 6, 8, -8], [4, 4, 6, -10]]]
[169, [[1, 2, 8, -10], [2, 4, -7, 10], [4, 4, 4, -11], [4, 5, 8, 8], [4, 6, 6, 9]]]
[170, [[0, 0, -7, 11], [0, -1, 5, 12], [0, 5, 8, -9], [-1, -3, -4, 12], [2, 2, 9, -9], [2, 3, 6, -11], [3, -4, 8, 9], [3, 5, 6, -10], [6, -6, 7, -7]]]
[171, [[0, -5, -5, 11], [1, 1, -5, 12], [3, -3, -3, 12], [3, 4, 5, -11], [3, 7, 7, 8], [4, 5, 7, 9]]]
[172, [[1, 1, 1, 13], [1, -1, -7, 11], [1, 3, 9, -9], [1, 5, 5, -11], [-5, 7, 7, 7]]]
[173, [[0, 2, -5, 12], [0, 3, 8, -10], [0, 4, -6, 11], [-1, 6, 6, -10], [-2, 3, -4, 12], [3, 6, 8, 8]]]
[174, [[0, 1, 2, 13], [0, -2, 7, 11], [1, -2, 5, 12], [1, -3, 8, 10], [1, 4, 6, -11], [-2, 5, -8, 9], [3, 4, 7, -10], [5, -6, -7, 8]]]
[175, [[-1, -1, -2, 13], [-1, -2, -7, 11], [1, 5, -7, 10], [-2, 3, 9, -9], [2, 5, 5, -11], [3, 3, 6, -11], [3, 6, 7, 9], [5, 5, 5, 10]]]
[176, [[0, -4, -4, 12], [2, 6, 6, -10]]]
[177, [[0, -2, -2, 13], [0, -7, 8, 8], [2, -3, -8, 10], [2, 4, 6, -11], [4, 4, 8, 9], [4, 5, 6, 10]]]
[178, [[0, 0, 3, 13], [0, -3, -5, 12], [0, 4, 9, -9], [-1, 2, 2, 13], [2, 2, 7, -11], [2, 5, 7, -10], [3, -3, 4, 12], [4, -4, 5, 11], [-4, 7, -7, 8], [5, -5, 8, 8], [-5, 6, 6, 9]]]
[179, [[0, -1, -3, 13], [0, -3, -7, 11], [0, 7, -7, 9], [-1, 3, -5, 12], [3, 5, 8, 9]]]
[180, [[1, -1, 3, 13], [1, 3, 7, -11], [-1, 7, 7, -9], [3, -3, 9, -9], [3, -5, -5, 11], [5, -5, 7, 9]]]
[181, [[0, 0, -9, 10], [-1, 4, 8, -10], [2, 7, 8, 8], [3, 6, 6, 10], [4, 4, 7, 10]]]
[182, [[0, 1, 9, -10], [0, 5, 6, -11], [-1, -1, 6, 12], [-1, 6, 8, -9], [2, -3, 5, 12], [2, -4, 9, 9], [3, 3, 8, -10], [3, -4, 6, 11], [4, -6, -7, 9]]]
[183, [[1, 1, 9, -10], [1, -2, -3, 13], [1, 5, 6, -11], [2, 3, 7, -11], [2, 7, 7, 9], [3, 5, 7, 10], [-6, 7, 7, -7]]]
[184, [[0, -2, 6, 12], [2, 4, 8, -10]]]
[185, [[0, 0, -4, 13], [0, 2, 9, -10], [0, 6, -7, 10], [1, 2, -6, 12], [2, 6, 8, 9], [-3, 4, -4, 12], [6, -6, -7, 8]]]
[186, [[0, -1, 4, 13], [0, 1, -8, 11], [0, 4, 7, -11], [1, 2, -9, 10], [1, -4, -5, 12], [1, 6, 7, -10], [2, -2, 3, 13], [-3, 7, 8, -8], [4, -5, 8, 9], [5, -5, 6, 10]]]
[187, [[1, -1, -4, 13], [-1, -1, -8, 11], [1, 4, 7, -11], [-3, -3, -5, 12], [3, 4, 9, 9], [4, 5, 5, 11], [-5, 7, 7, -8]]]
[188, [[-1, 5, 9, -9], [3, -3, -7, 11]]]
[189, [[0, -2, -8, 11], [0, 3, -6, 12], [2, -2, -9, 10], [-2, 4, -5, 12], [2, 6, 7, 10], [3, 4, 8, 10], [4, 4, 6, 11], [-5, 6, 8, -8], [6, 6, 6, -9]]]
[190, [[0, 3, -9, 10], [1, -2, 4, 13], [1, 2, 8, -11], [1, -3, 6, 12], [1, 5, 8, -10], [2, -4, 7, 11], [-3, 6, -8, 9], [4, -5, 7, 10]]]
[191, [[-1, 3, 9, -10], [2, -3, -3, 13], [2, 5, 9, 9], [3, 5, 6, 11], [5, 6, 7, -9]]]
[192, [[4, 4, 4, -12]]]
[193, [[0, 6, 6, -11], [1, 8, 8, 8], [2, -2, -4, 13], [2, 2, 8, -11], [-2, -3, -6, 12], [2, 5, 8, 10]]]
[194, [[0, -1, -7, 12], [0, 3, 8, -11], [1, -6, -6, 11], [2, 3, 9, -10], [3, 4, 5, -12], [3, -6, -7, 10], [4, -4, 9, -9]]]
[195, [[0, 1, -5, 13], [0, 5, 7, -11], [1, 3, 8, -11], [-1, -5, -5, 12], [1, 7, 8, 9], [3, 4, 7, 11], [4, 7, 7, -9], [5, 5, 8, -9]]]
[196, [[-1, -1, 5, 13], [1, -5, -7, 11], [3, 3, -3, 13], [3, -5, 9, 9], [5, 5, 5, -11], [7, 7, -7, -7]]]
[197, [[-1, 4, -6, 12], [2, 6, 6, 11], [4, 6, 8, -9]]]
[198, [[0, 1, 1, 14], [0, -2, 5, 13], [0, 7, 7, -10], [1, -2, -7, 12], [1, 4, 9, -10], [2, -3, 4, 13], [-2, 3, -8, 11], [2, 5, 5, -12], [-2, 7, 8, -9], [3, 3, 6, -12], [3, -5, 8, 10], [4, 5, 6, -11], [6, -7, -7, 8]]]
[199, [[-1, 2, -5, 13], [1, 6, 9, 9], [1, 7, 7, 10], [2, 5, 7, 11], [3, 3, 9, 10]]]
[200, [[0, 0, 2, 14], [0, 0, 10, -10], [0, 6, 8, -10], [2, 4, 6, -12], [6, -6, 8, 8]]]

160515(3)

Ruby


e とπ(2)

それぞれの分子について、
オンライン整数列大辞典の
A053557(http://oeis.org/A053557/list)、
A086116(http://oeis.org/A086116/list)
と比較し、答え合わせしてみる。
(注意:bn の分子がA086116 にたまたま一致しているだけかもしれません。)

def A053557(n)
  ary = [1, 0, 1]
  return ary[0..n] if n < 3
  a, b = 0, 1
  m = 2
  while m < n
    a, b = b, b + a / (m - 1r)
    ary << b.numerator
    m += 1
  end
  ary
end

def f(n)
  ary = [0, 1]
  return ary[0..n] if n < 2
  a, b = 0, 1
  m = 2
  while m < n
    a, b = b, a + b / (m - 1r)
    ary << b.numerator
    m += 1
  end
  ary
end

def A086116(n)
  f(n + 1)[1..-1]
end

ary = A053557(23)
# OEIS A053557のデータ
ary0 =
[1,0,1,1,3,11,53,103,2119,16687,16481,1468457,
 16019531,63633137,2467007773,34361893981,
 15549624751,8178130767479,138547156531409,
 92079694567171,4282366656425369,72289643288657479,
 6563440628747948887,39299278806015611311]
# 一致の確認
p ary == ary0

ary = A086116(33)
# OEIS A086116のデータ
ary0 =
[1,1,3,3,15,15,35,35,315,315,693,693,3003,3003,
 6435,6435,109395,109395,230945,230945,969969,
 969969,2028117,2028117,16900975,16900975,35102025,
 35102025,145422675,145422675,300540195,300540195,
 9917826435]
# 一致の確認
p ary == ary0

出力結果
true
true

160515(2)

Ruby


e とπ(1)

http://www004.upp.so-net.ne.jp/s_honma/mathbun/mathbun324.htm
に載っている漸化式が面白い。

a1 = 0,
a2 = 1,
a3 = a2 + a1 / 1 = 1 + 0 = 1,
a4 = a3 + a2 / 2 = 1 + 1 / 2 = 3 / 2,
a5 = a4 + a3 / 3 = 3 / 2 + 1 / 3 = 11 / 6,
a6 = a5 + a4 / 4 = 11 / 6 + 3 / 8 = 53 / 24,

に対し、n / an → e となる。
一方、
b1 = 0,
b2 = 1,
b3 = b2 / 1 + b1 = 1 + 0 = 1,
b4 = b3 / 2 + b2 = 1 / 2 + 1 = 3 /2,
b5 = b4 / 3 + b3 = 1 / 2 + 1 = 3 /2,
b6 = b5 / 4 + b4 = 3 / 8 + 3 / 2 = 15 / 8,

に対し、2n / (bn)^2 → π となる。

def a(n)
  ary = [0, 1]
  a, b = 0, 1
  m = 2
  while m < n
    a, b = b, b + a / (m - 1r)
    ary << b
    m += 1
  end
  ary
end

def b(n)
  ary = [0, 1]
  a, b = 0, 1
  m = 2
  while m < n
    a, b = b, a + b / (m - 1r)
    ary << b
    m += 1
  end
  ary
end

n0 = 100
p ary0 = a(n0)
p n0.to_f / ary0[-1]
n1 = 200
p ary1 = b(n1)
p (2 * n1).to_f / ary1[-1] ** 2

出力結果
[0, 1, (1/1), (3/2), (11/6), (53/24), (103/40), (2119/720), (16687/5040), (16481/4480), (1468457/362880), (16019531/3628800), (63633137/13305600), (2467007773/479001600), (34361893981/6227020800), (15549624751/2641766400), (8178130767479/1307674368000), (138547156531409/20922789888000), (92079694567171/13173608448000), (4282366656425369/582033973248000), (72289643288657479/9357315416064000), (6563440628747948887/810967336058880000), (39299278806015611311/4644631106519040000), (9923922230666898717143/1124000727777607680000), (79253545592131482810517/8617338912961658880000), (5934505493938805432851513/620448401733239439360000), (14006262966463963871240459/1410110003939180544000000), (461572649528573755888451011/44810162347400626176000000), (116167945043852116348068366947/10888869450418352160768000000), (3364864615063302680426807870189/304888344611713860501504000000), (277778998066291010992075323719/24357471057134165164032000000), 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2.718281828459045
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(140771089053929410103250840619832478385300791109703523041625/12554203470773361527671578846415332832204710888928069025792), (282964108300322753641888053367138012107826842735666677629125/25108406941546723055343157692830665664409421777856138051584)]
3.149456428081303

160515

Ruby


素数でしりとり

素数を最初から順に
2 → 23 → 31 → …
とつなげていく。
オンライン整数列大辞典の
https://oeis.org/A061448/b061448.txt
に途中まで載っているが、自分で求めてみた。
(ただし、素数を列挙するという非効率なやり方です。)

require 'prime'

cnt = 0
d = 2
ary = Prime.each(10 ** 7).to_a
ary.each{|a|
  str = a.to_s
  if d == str[0].to_i
    cnt += 1
    print cnt
    print ' '
    puts a
    d = str[-1].to_i
  end
}

出力結果
1 2
2 23
3 31
4 101
5 103
6 307
7 701
8 1009
9 9001
10 10007
11 70001
12 100003
13 300007
14 700001
15 1000003
16 3000017
17 7000003

2016年5月14日土曜日

160514

整数列のPROG の編集

https://oeis.org/A272890
のPROG を編集しました。

2016年5月11日水曜日

160511

整数列のLINKS の編集(15)

https://oeis.org/A100843
https://oeis.org/A100845
のLINKS を編集しました。

2016年5月10日火曜日

160510

整数列のLINKS の編集(14)

https://oeis.org/A272891
https://oeis.org/A072214
のLINKS を編集しました。

2016年5月6日金曜日

160506

Ruby


フィボナッチ数列でしりとり

フィボナッチ数列を最初から順に
1 → 13 → 34 → …
とつなげていく。
(ただし、0 が最後になればそこで終了。)
オンライン整数列大辞典の
https://oeis.org/A271750/b271750.txt
に答えが載っているが、自分で求めてみた。

cnt = 1
a, b = 1, 2
d = 1
puts '1 1'
while cnt < 100
  a, b = b, a + b
  str = a.to_s
  if d == str[0].to_i
    cnt += 1
    print cnt
    print ' '
    puts a
    d = str[-1].to_i
    break if d == 0
  end
end

出力結果
1 1
2 13
3 34
4 4181
5 10946
6 63245986
7 6557470319842
8 27777890035288
9 806515533049393
10 3416454622906707
11 7540113804746346429
12 927372692193078999176
13 6356306993006846248183
14 3311648143516982017180081
15 14028366653498915298923761
16 155576970220531065681649693
17 30960598847965113057878492344
18 42230279526998466217810220532898
19 8404037832974134882743767626780173
20 35600075545958458963222876581316753
21 394810887814999156320699623170776339
22 9663391306290450775010025392525829059713
23 3111581989804070186099320645726169127737705
24 5034645418285014325766435419644478339818233
25 34507973060837282187130139035400899082304280

2016年5月5日木曜日

160505(7)

整数列のLINKS の編集(13)

https://oeis.org/A272628
https://oeis.org/A272629
のLINKS を編集しました。

160505(6)

Ruby


Somos' sequence(2)

さらに調べてみた。

def b(k, n)
  b = Array.new(2 * k + 3, 1)
  (2 * k + 3..n).each{|i|
    j = (b[i - 1] * b[i - 2 * k - 2] + b[i - k - 1] * b[i - k - 2]) / b[i - 2 * k - 3].to_r
    j = j.to_i if j.denominator == 1
    b[i] = j
  }
  b[0..n]
end

(20..50).each{|i| p [i, b(i, 200)]}

出力結果
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