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%I #12 Jul 22 2019 06:30:17
%S 1,3,7,15,32,66,139,285,599,1227,2577,5277,11075,22671,47543,97287,
%T 203860,417006,873175,1785513,3736210,7637604,15972143,32641221,
%U 68224004,139389570,291199307,594818781,1242097912,2536656174
%N a(n) = Sum_{j=0..n} Sum_{i=0..n} T(j,i), T given by A026736.
%H Vaclav Kotesovec, <a href="/A026745/b026745.txt">Table of n, a(n) for n = 0..2000</a> (terms 0..1000 from G. C. Greubel)
%F a(n) ~ c * phi^(3*n/2), where c = 1/2 + 3*phi^2 / (2*sqrt(5)) if n is even, c = 3*phi^(5/2) / (2*sqrt(5)) if n is odd and phi = A001622 = (1+sqrt(5))/2 is the golden ratio. - _Vaclav Kotesovec_, Jul 22 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]]];
%t b[n_]:= Sum[T[n, j], {j,0,n}]; Table[Sum[b[j], {j,0,n}], {n,0,35}] (* _G. C. Greubel_, Jul 22 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 def b(n): return sum(T(n, j) for j in (0..n))
%o [sum(b(j) for j in (0..n)) for n in (0..35)] # _G. C. Greubel_, Jul 22 2019
%Y Cf. A026736.
%K nonn
%O 0,2
%A _Clark Kimberling_
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