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A376388 a(n) = P(n+1, n+1) where P(n, m) = P(n, m-1) + P(n-1, m + f(m-n)) for n < m with P(n, m) = P(n-1, m) for 0 <= m <= n and P(0, m) = 1 for m >= 0 where f(n) = [(n mod 5) > 0].

%I A376388 #8 Sep 22 2024 05:15:16
%S A376388 1,2,6,23,101,478,2367,12088,63166,336098,1814847,9920360,54789989,
%T A376388 305289034,1714103538,9688492693,55083466105,314806198628,
%U A376388 1807505286027,10421360638793,60311752073306,350235881381542,2040182863190499,11918253416566762,69805636091312473
%N A376388 a(n) = P(n+1, n+1) where P(n, m) = P(n, m-1) + P(n-1, m + f(m-n)) for n < m with P(n, m) = P(n-1, m) for 0 <= m <= n and P(0, m) = 1 for m >= 0 where f(n) = [(n mod 5) > 0].
%C A376388 Conjecture: cases f(n) = n mod 2 and f(n) = [(n mod 3) > 0] both gives A006318.
%H A376388 Vaclav Kotesovec, <a href="/A376388/b376388.txt">Table of n, a(n) for n = 0..1000</a>
%F A376388 From _Vaclav Kotesovec_, Sep 22 2024: (Start)
%F A376388 Recurrence: (n+1)*a(n) = (11*n-4)*a(n-1) - 12*(3*n-5)*a(n-2) + 3*(13*n-42)*a(n-3) + 4*(n+7)*a(n-4) - 3*(7*n-24)*a(n-5) + (n-5)*a(n-6).
%F A376388 a(n) ~ sqrt(114 - 63*sqrt(3) + sqrt(33*(795 - 412*sqrt(3)))) * (5 + 2*sqrt(3) + sqrt(9 + 4*sqrt(3)))^n / (sqrt(Pi) * n^(3/2) * 2^(n + 5/2)). (End)
%o A376388 (PARI) upto(n) = my(v1); v1 = vector(2*(n+1), i, 1); v2 = vector(n+1, i, 0); v2[1] = 1; for(i=1, n, for(j=i+1, 2*(n+1)-i, v1[j] = v1[j+(((j-i)%5)>0)] + v1[j-1]); v2[i+1] = v1[i+1]); v2
%Y A376388 Cf. A006318.
%K A376388 nonn
%O A376388 0,2
%A A376388 _Mikhail Kurkov_, Sep 22 2024