A026735 a(n) = Sum_{k=0..floor(n/2)} T(n-k,k), T given by A026725.

1, 1, 2, 3, 6, 9, 15, 28, 43, 71, 130, 201, 331, 597, 928, 1525, 2720, 4245, 6965, 12315, 19280, 31595, 55472, 87067, 142539, 248802, 391341, 640143, 1111864, 1752007, 2863871, 4953162, 7817033, 12770195, 22004810, 34775005, 56779815

Offset: 0

Author

Clark Kimberling

Keywords

• nonn

Links

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

Programs

GAP

T:= function(n,k)
    if n<0 then return 0;
    elif k=0 or k=n then return 1;
    elif 2*k=n-1 then return T(n-1, k-1) + T(n-2, k-1) + T(n-1, k);
    else return T(n-1, k-1) + T(n-1, k);
    fi;
  end;
List([0..30], n-> Sum([0..Int(n/2)], k-> T(n-k,k) )); # G. C. Greubel, Oct 26 2019

Maple

A026725:= proc(n,k) option remember;
    if n<0 or k<0 then 0;
    elif k=0 or k=n then 1;
    elif 2*k = n-1 then procname(n-1,k-1)+procname(n-2,k-1) + procname(n-1,k) ;
   else procname(n-1,k-1)+procname(n-1,k) ;
   fi;
end proc:
seq(add(A026725(n-k,k), k=0..floor(n/2)), n=0..30); # G. C. Greubel, Oct 26 2019

Mathematica

T[n_, k_]:= T[n, k]= If[n<0, 0, If[k==0 || k==n, 1, If[OddQ[n] && k==(n-1)/2, T[n-1, k-1] + T[n-2, k-1] + T[n-1, k], T[n-1, k-1] + T[n-1, k]]]]; Table[Sum[T[n-k, k], {k,0,Floor[n/2]}], {n,0,30}] (* G. C. Greubel, Oct 26 2019 *)

PARI

T(n,k) = if(n<0, 0, if(k==n || k==0, 1, if(2*k==n-1, T(n-1, k-1) + T(n-2, k-1) + T(n-1, k), T(n-1, k-1) + T(n-1, k) )));
vector(31, n, sum(j=0,(n-1)\2, T(n-j,j)) ) \\ G. C. Greubel, Oct 26 2019

Sage

@CachedFunction
def T(n, k):
    if (n<0): return 0
    elif (k==0 or k==n): return 1
    elif (mod(n,2)==0 and k==(n-1)/2): return T(n-1, k-1) + T(n-2, k-1) + T(n-1, k)
    else: return T(n-1, k-1) + T(n-1, k)
[sum( T(n-j, j) for j in (0..floor(n/2))) for n in (0..30)] # G. C. Greubel, Oct 26 2019

Formula

Conjecture: G.f.:-1/2*(2*x^6-5*x^4+8*x^3+x-2+x*(x-1)*(x^2+x+1)*(1-4*x^3)^(1/2))/(x^6+4*x^3-1)/(x^2+x-1). - Maksym Voznyy (voznyy(AT)mail.ru), Aug 11 2009