A114213 -id:A114213 - cp's OEIS frontend

A114214 Diagonal sums of number triangle A114213.

1, 1, 2, 2, 3, 3, 3, 3, 4, 4, 5, 5, 5, 5, 4, 4, 5, 5, 7, 7, 8, 8, 7, 7, 7, 7, 8, 8, 7, 7, 5, 5, 6, 6, 9, 9, 11, 11, 10, 10, 11, 11, 13, 13, 12, 12, 9, 9, 9, 9, 12, 12, 13, 13, 11, 11, 10, 10, 11, 11, 9, 9, 6, 6, 7, 7, 11, 11, 14, 14, 13, 13, 15, 15, 18, 18, 17, 17, 13, 13, 14, 14, 19, 19, 21

Offset: 0

Paul Barry, Nov 17 2005

Conjecture: a(n) = A007306(floor(n/2)+1). - Georg Fischer, Nov 28 2022

Georg Fischer, Table of n, a(n) for n = 0..1000
Jeffrey Shallit and Lukas Spiegelhofer, Continuants, run lengths, and Barry's modified Pascal triangle, arXiv:1710.06203 [math.CO], 2017.

Cf. A007306, A114213.

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

a(n) = Sum_{k=0..floor(n/2)} mod(Sum_{j=0..n-2k} C(k, j) C(n-2k, j) (1+(-1)^j)/2, 2). (corrected by Jeffrey Shallit, May 18 2016)

A114212 Generalized Gould sequence.

Original entry on oeis.org

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

Paul Barry, Nov 17 2005

Row sums of A114213.

			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)

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

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

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

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.

cp's OEIS Frontend

A114214 Diagonal sums of number triangle A114213.

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