A167667 -id:A167667 - cp's OEIS frontend

A308663 Partial sums of A097805.

1, 1, 2, 2, 3, 4, 4, 5, 7, 8, 8, 9, 12, 15, 16, 16, 17, 21, 27, 31, 32, 32, 33, 38, 48, 58, 63, 64, 64, 65, 71, 86, 106, 121, 127, 128, 128, 129, 136, 157, 192, 227, 248, 255, 256, 256, 257, 265, 293, 349, 419, 475, 503, 511, 512

Offset: 0

Paul Curtz, Jun 15 2019

Curtz (1965), page 15, from right to left, gives (F1):
  1/2;
  1/4,  3/4;
  1/8,  4/8,   7/8;
  1/16, 5/16, 11/16, 15/16;
  ... .
Numerators + Denominators = (C) =
   3;
   5,  7;
   9, 12, 15;
  17, 21, 27, 31;
  ... .
This is the current sequence without powers of 2.
The triangle (P) for a(n) is
  1;
  1, 2;
  2, 3,  4;
  4, 5,  7,  8;
  8, 9, 12, 15, 16;
  ... .
(C) is the core of (P).
Extension of (F1). (F2) =
  0/1;
  0/1, 1/1;
  0/2, 1/2, 2/2;
  0/4, 1/4, 3/4, 4/4;
  0/8, 1/8, 4/8, 7/8, 8/8;
  ... .
(Mentioned, without 0's, op. cit., page 16.)
a(n) = Numerators + Denominators.
Row sums of triangle (P): A084858(n).
From right to left, with alternating signs: 1, 1, 3, 2, 12, 8, 48, 32, ..., see A098646.
For triangle (C), row sums give A167667(n+1).
From right to left, with alternating signs: A098646(n).
Rank of A016116(n): 0 together with A117142.

Paul Curtz, Accélération de la convergence de certaines séries alternées à l'aide des fonctions de sommation, Thèse de 3ème Cycle d'Analyse Numérique, Faculté des Sciences de l'Université de Paris, 4 mai 1965.

Cf. A097805.

T(n,k) = ceiling(2^(n-1)) + Sum_{j=0..k-1} binomial(n-1,j). - Alois P. Heinz, Jun 15 2019

a(n+1) = a(n) + A097805(n+1) for n >= 0.

Edited by N. J. A. Sloane, Sep 15 2019

A384694 Sum of the number of cells alive after 2 generations of Conway's game of life for initial 1 X n cells taken in all 2^n combinations of alive or dead.

Original entry on oeis.org

0, 0, 3, 12, 35, 92, 228, 544, 1264, 2880, 6464, 14336, 31488, 68608, 148480, 319488, 684032, 1458176, 3096576, 6553600, 13828096, 29097984, 61079552, 127926272, 267386880, 557842432, 1161822208, 2415919104, 5016387584, 10401873920, 21541945344, 44560285696, 92073361408, 190052302848, 391915765760

Offset: 0

SiYang Hu, Jun 07 2025

			For n = 5, there are 5 ways for the cells to evolve into a blinker: ..OOO, O.OOO, .OOO., OOO.., OOO.O; 4 ways for the cells to evolve into a beehive predecessor and then a beehive: OOOO., .OOOO; 1 way for it to evolve into 8 cells: OOOOO, so a(5) = 3 * 5 + 6 * 2 + 8 * 1 = 35.

LifeWiki, One cell thick pattern
Index entries for linear recurrences with constant coefficients, signature (4,-4).

Cf. A167667 (after one generation).

G.f.: x^2*(3 - x^2)/(1 - 4*x + 4*x^2).
a(n) = 2^(n - 5) * (11*n - 20).
E.g.f.: (9 - 4*x - 2*x^2 + exp(2*x)*(22*x - 9))/16. - Stefano Spezia, Jun 07 2025

cp's OEIS Frontend

A308663 Partial sums of A097805.

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