A160999 Row sums of A027052.

1, 2, 5, 12, 31, 84, 233, 656, 1865, 5338, 15355, 44342, 128455, 373100, 1086087, 3167634, 9254009, 27074666, 79316491, 232633206, 683026535, 2007327660, 5904415195, 17381265934, 51203990457, 150945252394, 445252685313

Offset: 0

Author

R. J. Mathar, Jun 01 2009

Keywords

• easy
• nonn

Examples

			a(2) = 1+0+1+2+1 = 5.
a(3) = 1+0+1+2+3+4+1 = 12.
		

Links

• G. C. Greubel, Table of n, a(n) for n = 0..1000

Programs

Maple

A027052 := proc(n,k) option remember; if k =0 or k = 2*n then 1; elif k = 1 then 0; elif k =2 then 1; else procname(n-1,k-3)+procname(n-1,k-2)+procname(n-1,k-1) ; fi; end:
A160999 := proc(n) add( A027052(n,k),k=0..2*n) ; end: seq(A160999(n),n=0..30) ;

Mathematica

T[n_, k_]:= T[n, k]= If[k==0 || k==2 || k==2*n, 1, If[k==1, 0, Sum[T[n-1, k-j], {j, 3}]]]; Table[Sum[T[n, k], {k,0,2*n}], {n,0,30}] (* G. C. Greubel, Nov 06 2019 *)

Sage

@CachedFunction
def T(n, k):
    if (k==0 or k==2 or k==2*n): return 1
    elif (k==1): return 0
    else: return sum(T(n-1, k-j) for j in (1..3))
[sum(T(n, k) for k in (0..2*n)) for n in (0..30)] # G. C. Greubel, Nov 06 2019

Formula

a(n) = Sum_{k=0..2*n} A027052(n,k).

Conjecture: (-n+2)*a(n) +(6*n-11)*a(n-1) +(-7*n+1)*a(n-2) +2*(-4*n+27)*a(n-3) +(5*n-28)*a(n-4) +(2*n-3)*a(n-5) +3*(n-5)*a(n-6)=0. - R. J. Mathar, May 26 2016