A344576 a(n) = f(n,n) where f(0,n) = f(n,0) = Fibonacci(n) and f(m,n) = f(m-1,n) + f(m,n-1) + f(m-1,n-1).

0, 2, 10, 52, 278, 1510, 8288, 45834, 254922, 1424252, 7986550, 44921582, 253320352, 1431678194, 8106897418, 45982821860, 261206625526, 1485765938390, 8461264982176, 48237937154554, 275275548126890, 1572297656021292, 8987888015996790, 51417128080562142

Offset

0,2

Comments

a(n+1)/a(n) tends to A156035.

Links

Table of n, a(n) for n=0..23.

Formula

a(n) = 2*Sum_{k=1..n} Fibonacci(k)*(A008288(n-1,n-k) + A008288(n-1,n-k-1)). - Andrew Howroyd, May 29 2021

G.f.: x*(3*x^2-18*x+3-(x+1)*sqrt(x^2-6*x+1))/((x^2-7*x+1)*(x^2-6*x+1)). - Alois P. Heinz, May 29 2021

a(n) = ((79-97*n+26*n^2)*a(n-1) + (-9+9*n-2*n^2)*a(n-4) + (107-111*n+26*n^2)*a(n-3) + (-322+352*n-88*n^2)*a(n-2)) / (5-7*n+2*n^2) for n >= 4. - José María Grau Ribas, Jun 19 2021

Mathematica

F[0, 0] = 0; F[m_, 0] := Fibonacci[m]; F[0, n_] := Fibonacci[n];
F[m_, n_] := F[m, n] =   F[m - 1 , n ] + F[m , n - 1] +  F[m - 1, n - 1];
Table[F[n, n], {n, 0, 100}]

Prog

(PARI) \\ here D(n, k) is A008288(n, k).
D(n, k) = {sum(d = 0, min(n, k), binomial(k, d)*binomial(n+k-d, k))}
a(n) = {2*sum(k=1, n, fibonacci(k)*(D(n-1, n-k) + D(n-1, n-k-1)))} \\ Andrew Howroyd, May 29 2021

Crossrefs

Cf. A000045, A001850, A008288, A156035.

Sequence in context: A075436 A319325 A361805 * A360317 A074612 A104497

Adjacent sequences: A344573 A344574 A344575 * A344577 A344578 A344579

Keyword

nonn

Author

José María Grau Ribas, May 24 2021

Status

approved