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%I #9 Dec 19 2021 04:29:49
%S 1,2,6,11,33,64,191,376,1122,2222,6636,13180,39395,78373,234414,
%T 466840,1397034,2784266,8335242,16620976,49773018,99291358,297406884,
%U 593484440,1777995535,3548969075,10633840743,21230215328,63620551947
%N a(n) = Sum_{j=0..n} T(n, j), where T is given by A026552.
%H G. C. Greubel, <a href="/A026564/b026564.txt">Table of n, a(n) for n = 0..1000</a>
%F a(n) = Sum_{j=0..n} A026552(n, j).
%t T[n_, k_]:= T[n, k]= If[k==0 || k==2*n, 1, If[k==1 || k==2*n-1, Floor[(n+2)/2], If[EvenQ[n], T[n-1, k-2] + T[n-1, k] + T[n-1, k-1], T[n-1, k-2] + T[n-1, k]]]]; (* T=A026552 *)
%t a[n_]:= a[n]= Block[{$RecursionLimit = Infinity}, Sum[T[n, k], {k,0,n}]];
%t Table[a[n], {n,0,40}] (* _G. C. Greubel_, Dec 19 2021 *)
%o (Sage)
%o @CachedFunction
%o def T(n,k): # T = A026552
%o if (k==0 or k==2*n): return 1
%o elif (k==1 or k==2*n-1): return (n+2)//2
%o elif (n%2==0): return T(n-1, k) + T(n-1, k-1) + T(n-1, k-2)
%o else: return T(n-1, k) + T(n-1, k-2)
%o @CachedFunction
%o def a(n): return sum( T(n,k) for k in (0..n) )
%o [a(n) for n in (0..40)] # _G. C. Greubel_, Dec 19 2021
%Y Cf. A026552, A026553, A026554, A026555, A026556, A026557, A026558, A026559, A026560, A026563, A026566, A026567, A027272, A027273, A027274, A027275, A027276.
%K nonn
%O 0,2
%A _Clark Kimberling_
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