cp's OEIS Frontend

A114212 Generalized Gould sequence.

%I A114212 #15 Jun 06 2021 09:05:14
%S A114212 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,
%T A114212 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,
%U A114212 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
%N A114212 Generalized Gould sequence.
%C A114212 Row sums of A114213.
%H A114212 Jeffrey Shallit and Lukas Spiegelhofer, <a href="https://arxiv.org/abs/1710.06203">Continuants, run lengths, and Barry's modified Pascal triangle</a>, arXiv:1710.06203 [math.CO], 2017.
%F A114212 a(n) = Sum_{k=0..n} (Sum_{j=0..n-k} C(k, j)*C(n-k, j)*((1 + (-1)^k)/2) mod 2).
%F A114212 a(n) = A001316(n) + A001316((n-2)/2)*(1 + (-1)^n)/2.
%e A114212 From _Omar E. Pol_, Jun 09 2009: (Start)
%e A114212 Triangle begins:
%e A114212   1;
%e A114212   2,3;
%e A114212   4,4,4,6;
%e A114212   8,6,4,6,8,8,8,12;
%e A114212   16,10,4,6,8,8,8,12,16,12,8,12,16,16,16,24;
%e A114212   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,...
%e A114212 Also, we can write the initial term followed by a triangle:
%e A114212   1;
%e A114212   2;
%e A114212   3,4;
%e A114212   4,4,6,8;
%e A114212   6,4,6,8,8,8,12,16;
%e A114212   10,4,6,8,8,8,12,16,12,8,12,16,16,16,24,32;
%e A114212   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,...
%e A114212 Also, we can write first two terms followed by a triangle:
%e A114212   1;
%e A114212   2;
%e A114212   3;
%e A114212   4,4;
%e A114212   4,6,8,6;
%e A114212   4,6,8,8,8,12,16,10;
%e A114212   4,6,8,8,8,12,16,12,8,12,16,16,16,24,32,18;
%e A114212   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,...
%e A114212 (End)
%o A114212 (PARI) T(n,k) = sum(j=0, n-k, binomial(k, j)*binomial(n-k, j)*(1+(-1)^j)/2) % 2; \\ A114213
%o A114212 a(n) = sum(k=0, n-1, T(n,k)); \\ _Michel Marcus_, Jun 06 2021
%Y A114212 Cf. A000079. [_Omar E. Pol_, Jun 09 2009]
%K A114212 easy,nonn
%O A114212 0,2
%A A114212 _Paul Barry_, Nov 17 2005