cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

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A144545 a(n) = 2^(n*(n-1))*(2^n + 1)*Product_{i=1..n-1} (4^i - 1).

Original entry on oeis.org

2, 3, 60, 25920, 197406720, 25015379558400, 51615733565620224000, 1718194449153210615595008000, 918817155086936330770931156779008000, 7877103854727828347931810809383874168094720000, 1081561598265935342583934931877242782978883444539392000000
Offset: 0

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Author

N. J. A. Sloane, Dec 30 2008

Keywords

Links

Crossrefs

Cf. A003923, A001308, A003053, A100221.

Programs

  • Maple
    g:=m->2^(m*(m-1))*mul( 4^i-1, i=1..m-1)*(2^m+1);
  • Mathematica
    a[n_] := 2^(n*(n-1))*(2^n + 1) * Product[4^i - 1, {i, 1, n-1}]; Array[a, 10, 0] (* Amiram Eldar, Jul 07 2025 *)
  • Python
    from math import prod
def A144545(n): return ((1<Chai Wah Wu, Jun 20 2022

Formula

a(n) ~ c * 2^(2*n^2-n), where c = A100221. - Amiram Eldar, Jul 07 2025

A330863 Decimal expansion of Product_{k>=1} (1 + 1/(-2)^k).

Original entry on oeis.org

5, 6, 8, 6, 9, 8, 9, 4, 6, 2, 6, 5, 4, 2, 8, 5, 0, 5, 9, 5, 4, 9, 7, 6, 7, 3, 7, 0, 7, 4, 4, 4, 4, 6, 5, 4, 2, 9, 0, 8, 5, 2, 4, 5, 1, 3, 8, 9, 3, 5, 9, 0, 2, 9, 3, 1, 9, 3, 4, 4, 0, 4, 6, 0, 1, 8, 3, 5, 3, 5, 6, 3, 2, 3, 0, 9, 1, 2, 6, 4, 0, 9, 6, 1, 4, 6, 4, 4, 1, 1, 7, 3, 0, 6, 1, 4, 8, 6, 0, 4, 8, 0, 2, 7, 2, 6, 9, 4, 1, 8
Offset: 0

Views

Author

Ilya Gutkovskiy, Apr 28 2020

Keywords

Examples

			(1 - 1/2) * (1 + 1/2^2) * (1 - 1/2^3) * (1 + 1/2^4) * (1 - 1/2^5) * ... = 0.568698946265428505954976737...

Crossrefs

Cf. A000593, A027637, A048651, A062510, A065446, A079555, A081845, A085011, A100221, A132020, A330862, A216206.

Programs

  • Mathematica
    RealDigits[QPochhammer[-1, -1/2]/2, 10, 110] [[1]]
N[3/QPochhammer[-2, 1/4], 120] (* Vaclav Kotesovec, Apr 28 2020 *)
  • PARI
    prodinf(k=1, 1 + 1/(-2)^k) \\ Michel Marcus, Apr 28 2020

Formula

Equals Product_{k>=1} 1/(1 + 1/2^(2*k-1)).
Equals exp(Sum_{k>=1} A000593(k)/(k*(-2)^k)).
From Peter Bala, Dec 15 2020: (Start)
Constant C = (2/3) - (1/3)*Sum_{n >= 0} (-1)^n * 2^(n^2)/( Product_{k = 1..n+1} 4^k - 1 ).
C = Sum_{n >= 0} 1/( Product_{k = 1..n} (-2)^k - 1 ) = 1 - 1/3 - 1/9 + 1/81 + 1/1215 - - + + ... = Sum_{n >= 0} 1/A216206(n).
C = 1 + Sum_{n >= 0} (-1/2)^(n+1)*Product_{k = 1..n} (1 + (-1/2)^k).
3*C = 2 - Sum_{n >= 0} (1/4)^(n+1)*Product_{k = 1..n} (1 + (-1/2)^k).
9*C = 5 - Sum_{n >= 0} (-1/8)^(n+1)*Product_{k = 1..n} (1 + (-1/2)^k).
81*C = 46 + Sum_{n >= 0} (1/16)^(n+1)*Product_{k = 1..n} (1 + (-1/2)^k).
1215*C = 691 + Sum_{n >= 0} (-1/32)^(n+1)*Product_{k = 1..n} (1 + (-1/2)^k).
The sequence [1, 2, 5, 46, 691, ...] is the sequence of numerators of the partial sums of the series Sum_{n >= 0} 1/A216206(n). (End)
Previous Showing 31-32 of 32 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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