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

%I A026735 #9 Oct 26 2019 16:15:29
%S A026735 1,1,2,3,6,9,15,28,43,71,130,201,331,597,928,1525,2720,4245,6965,
%T A026735 12315,19280,31595,55472,87067,142539,248802,391341,640143,1111864,
%U A026735 1752007,2863871,4953162,7817033,12770195,22004810,34775005,56779815
%N A026735 a(n) = Sum_{k=0..floor(n/2)} T(n-k,k), T given by A026725.
%H A026735 G. C. Greubel, <a href="/A026735/b026735.txt">Table of n, a(n) for n = 0..1000</a>
%F A026735 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
%p A026735 A026725:= proc(n,k) option remember;
%p A026735     if n<0 or k<0 then 0;
%p A026735     elif k=0 or k=n then 1;
%p A026735     elif 2*k = n-1 then procname(n-1,k-1)+procname(n-2,k-1) + procname(n-1,k) ;
%p A026735    else procname(n-1,k-1)+procname(n-1,k) ;
%p A026735    fi;
%p A026735 end proc:
%p A026735 seq(add(A026725(n-k,k), k=0..floor(n/2)), n=0..30); # _G. C. Greubel_, Oct 26 2019
%t A026735 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 *)
%o A026735 (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) )));
%o A026735 vector(31, n, sum(j=0,(n-1)\2, T(n-j,j)) ) \\ _G. C. Greubel_, Oct 26 2019
%o A026735 (Sage)
%o A026735 @CachedFunction
%o A026735 def T(n, k):
%o A026735     if (n<0): return 0
%o A026735     elif (k==0 or k==n): return 1
%o A026735     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)
%o A026735     else: return T(n-1, k-1) + T(n-1, k)
%o A026735 [sum( T(n-j, j) for j in (0..floor(n/2))) for n in (0..30)] # _G. C. Greubel_, Oct 26 2019
%o A026735 (GAP)
%o A026735 T:= function(n,k)
%o A026735     if n<0 then return 0;
%o A026735     elif k=0 or k=n then return 1;
%o A026735     elif 2*k=n-1 then return T(n-1, k-1) + T(n-2, k-1) + T(n-1, k);
%o A026735     else return T(n-1, k-1) + T(n-1, k);
%o A026735     fi;
%o A026735   end;
%o A026735 List([0..30], n-> Sum([0..Int(n/2)], k-> T(n-k,k) )); # _G. C. Greubel_, Oct 26 2019
%K A026735 nonn
%O A026735 0,3
%A A026735 _Clark Kimberling_