login
A114212
Generalized Gould sequence.
1
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
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
KEYWORD
easy,nonn
AUTHOR
Paul Barry, Nov 17 2005
STATUS
approved