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A026746 a(n) = Sum_{k=0..floor(n/2)} T(n-k,k), T given by A026736.

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