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

Author

Paul Barry, Nov 17 2005

Keywords

• easy
• nonn

Comments

Row sums of A114213.

Examples

			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)
		

Links

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

Crossrefs

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

Programs

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

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.