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%I #11 Jul 20 2019 01:16:29
%S 1,1,2,3,5,9,14,23,42,65,107,194,301,495,890,1385,2275,4058,6333,
%T 10391,18404,28795,47199,83079,130278,213357,373512,586869,960381,
%U 1673271,2633652,4306923,7472326,11779249,19251575,33275451,52527026
%N a(n) = Sum_{k=0..floor(n/2)} T(n-k,k), T given by A026736.
%H G. C. Greubel, <a href="/A026746/b026746.txt">Table of n, a(n) for n = 0..1000</a>
%F a(n) ~ n * phi^(n-2) / 15, where phi = A001622 = (1+sqrt(5))/2 is the golden ratio. - _Vaclav Kotesovec_, Jul 19 2019
%t T[n_, k_]:= T[n, k] = If[k==0 || k==n, 1, If[EvenQ[n] && k==(n-2)/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, 40}] (* _G. C. Greubel_, Jul 19 2019 *)
%o (PARI)
%o T(n,k) = if(k==n || k==0, 1, k==n-1, n, if((n%2)==0 && k==(n-2)/2, T(n-1, k-1) + T(n-2, k-1) + T(n-1, k), T(n-1, k-1) + T(n-1, k) ));
%o vector(30, n, n--; sum(k=0, n\2, T(n-k,k))) \\ _G. C. Greubel_, Jul 19 2019
%o (Sage)
%o @CachedFunction
%o def T(n, k):
%o if (k==0 or k==n): return 1
%o elif (mod(n,2)==0 and k==(n-2)/2): return T(n-1, k-1) + T(n-2, k-1) + T(n-1, k)
%o else: return T(n-1, k-1) + T(n-1, k)
%o [sum(T(n-k,k) for k in (0..floor(n/2))) for n in (0..40)] # _G. C. Greubel_, Jul 19 2019
%o (GAP)
%o T:= function(n,k)
%o if k=0 or k=n then return 1;
%o elif (n mod 2)=0 and k=Int((n-2)/2) then return T(n-1, k-1) + T(n-2, k-1) + T(n-1, k);
%o else return T(n-1, k-1) + T(n-1, k);
%o fi;
%o end;
%o List([0..20], n-> Sum([0..Int(n/2)], k-> T(n-k,k) )); # _G. C. Greubel_, Jul 19 2019
%Y Cf. A026736.
%K nonn
%O 0,3
%A _Clark Kimberling_
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