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

Author

Paul Barry, Nov 17 2005

Keywords

• easy
• nonn

Comments

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

Links

• 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.

Crossrefs

Cf. A007306, A114213.

Programs

PARI

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

Formula

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)