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

0,3

Links

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

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

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 *)

Prog

(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
(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

Crossrefs

Sequence in context: A293631 A018158 A057928 * A006868 A067435 A035494

Adjacent sequences: A026732 A026733 A026734 * A026736 A026737 A026738

Keyword

nonn

Author

Clark Kimberling

Status

approved