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].

1, 2, 6, 23, 101, 478, 2367, 12088, 63166, 336098, 1814847, 9920360, 54789989, 305289034, 1714103538, 9688492693, 55083466105, 314806198628, 1807505286027, 10421360638793, 60311752073306, 350235881381542, 2040182863190499, 11918253416566762, 69805636091312473

Offset

0,2

Comments

Conjecture: cases f(n) = n mod 2 and f(n) = [(n mod 3) > 0] both gives A006318.

Links

Vaclav Kotesovec, Table of n, a(n) for n = 0..1000

Formula

From Vaclav Kotesovec, Sep 22 2024: (Start)

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).

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)

Prog

(PARI) list(n) = my(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

Crossrefs

Cf. A006318.

Sequence in context: A387667 A193038 A213090 * A218225 A279572 A263576

Adjacent sequences: A376385 A376386 A376387 * A376389 A376390 A376391

Keyword

nonn

Author

Mikhail Kurkov, Sep 22 2024

Status

approved