A114212 Generalized Gould sequence.

1, 2, 3, 4, 4, 4, 6, 8, 6, 4, 6, 8, 8, 8, 12, 16, 10, 4, 6, 8, 8, 8, 12, 16, 12, 8, 12, 16, 16, 16, 24, 32, 18, 4, 6, 8, 8, 8, 12, 16, 12, 8, 12, 16, 16, 16, 24, 32, 20, 8, 12, 16, 16, 16, 24, 32, 24, 16, 24, 32, 32, 32, 48, 64, 34, 4, 6, 8, 8, 8, 12, 16, 12, 8, 12, 16, 16, 16, 24, 32, 20, 8

Offset

0,2

Comments

Row sums of A114213.

Links

Table of n, a(n) for n=0..81.

Jeffrey Shallit and Lukas Spiegelhofer, Continuants, run lengths, and Barry's modified Pascal triangle, arXiv:1710.06203 [math.CO], 2017.

Formula

a(n) = Sum_{k=0..n} (Sum_{j=0..n-k} C(k, j)*C(n-k, j)*((1 + (-1)^k)/2) mod 2).

a(n) = A001316(n) + A001316((n-2)/2)*(1 + (-1)^n)/2.

Example

From Omar E. Pol, Jun 09 2009: (Start)
Triangle begins:
  1;
  2,3;
  4,4,4,6;
  8,6,4,6,8,8,8,12;
  16,10,4,6,8,8,8,12,16,12,8,12,16,16,16,24;
  32,18,4,6,8,8,8,12,16,12,8,12,16,16,16,24,32,20,8,12,16,16,16,24,32,24,...
Also, we can write the initial term followed by a triangle:
  1;
  2;
  3,4;
  4,4,6,8;
  6,4,6,8,8,8,12,16;
  10,4,6,8,8,8,12,16,12,8,12,16,16,16,24,32;
  18,4,6,8,8,8,12,16,12,8,12,16,16,16,24,32,20,8,12,16,16,16,24,32,24,16,...
Also, we can write first two terms followed by a triangle:
  1;
  2;
  3;
  4,4;
  4,6,8,6;
  4,6,8,8,8,12,16,10;
  4,6,8,8,8,12,16,12,8,12,16,16,16,24,32,18;
  4,6,8,8,8,12,16,12,8,12,16,16,16,24,32,20,8,12,16,16,16,24,32,24,16,24,32,...
(End)

Prog

(PARI) T(n, k) = sum(j=0, n-k, binomial(k, j)*binomial(n-k, j)*(1+(-1)^j)/2) % 2; \\ A114213
a(n) = sum(k=0, n-1, T(n, k)); \\ Michel Marcus, Jun 06 2021

Crossrefs

Cf. A000079. [Omar E. Pol, Jun 09 2009]

Sequence in context: A131798 A206925 A339363 * A344351 A108355 A346049

Adjacent sequences: A114209 A114210 A114211 * A114213 A114214 A114215

Keyword

easy,nonn

Author

Paul Barry, Nov 17 2005

Status

approved