A055837 T(2n+3,n), where T is the array in A055830.

3, 15, 73, 361, 1806, 9122, 46425, 237721, 1223365, 6321965, 32784830, 170528190, 889291352, 4648068192, 24342384337, 127707864849, 671047979300, 3531026714720, 18603737992455, 98129545962855, 518149580437560

Offset: 0

Author

Clark Kimberling, May 28 2000

Keywords

• nonn

Links

• G. C. Greubel, Table of n, a(n) for n = 0..500

Crossrefs

Cf. A055830.

Programs

Maple

with(combinat);
T:= proc(n, k) option remember;
      if k<0 or k>n then 0
    elif k=0 then fibonacci(n+1)
    elif n=1 and k=1 then 0
    else T(n-1, k-1) + T(n-1, k) + T(n-2, k)
      fi; end:
seq(T(2*n+3, n), n=0..30); # G. C. Greubel, Jan 21 2020

Mathematica

T[n_, k_]:= T[n, k]= If[k<0 || k>n, 0, If[k==0, Fibonacci[n+1], If[n==1 && k==1, 0, T[n-1, k-1] + T[n-1, k] + T[n-2, k]]]]; Table[T[2*n+3, n], {n,0,30}] (* G. C. Greubel, Jan 21 2020 *)

Sage

@CachedFunction
def T(n, k):
    if (k<0 or k>n): return 0
    elif (k==0): return fibonacci(n+1)
    elif (n==1 and k==1): return 0
    else: return T(n-1, k-1) + T(n-1, k) + T(n-2, k)
[T(2*n+3, n) for n in (0..30)] # G. C. Greubel, Jan 21 2020

Formula

Conjecture: 5*n*(n+2)*(11*n-4)*a(n) +(-242*n^3-330*n^2+29*n+42)*a(n-1) -3*(3*n-1)*(11*n+7)*(3*n-2)*a(n-2)=0. - R. J. Mathar, Mar 13 2016