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A036256 a(n) = Sum_{i=0..n} binomial(i,floor(i/2)).

%I A036256 #36 Feb 16 2025 08:32:37
%S A036256 1,2,4,7,13,23,43,78,148,274,526,988,1912,3628,7060,13495,26365,50675,
%T A036256 99295,191673,376429,729145,1434577,2786655,5490811,10691111,21091711,
%U A036256 41150011,81266611,158825371,313942891,614483086,1215563476
%N A036256 a(n) = Sum_{i=0..n} binomial(i,floor(i/2)).
%C A036256 Equals row sums of triangle A145972. - _Gary W. Adamson_, Oct 25 2008
%C A036256 a(n-1) is the graph bandwidth of the n-hypercube graph Q_n. - _Eric W. Weisstein_, Jul 12 2011
%H A036256 G. C. Greubel, <a href="/A036256/b036256.txt">Table of n, a(n) for n = 0..1000</a>
%H A036256 L. H. Harper, <a href="https://doi.org/10.1016/S0021-9800(66)80059-5">Optimal numberings and isoperimetric problems on graphs</a>, J. Comb. Th. 1 (1966), 385-393.
%H A036256 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/GraphBandwidth.html">Graph Bandwidth</a>
%H A036256 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/HypercubeGraph.html">Hypercube Graph</a>
%F A036256 G.f.: 2/((1-z)*(1-2*z+sqrt(1-4*z^2))). - _Emeric Deutsch_, Nov 25 2003
%F A036256 a(n) ~ 2^(n+3/2) / sqrt(Pi*n) * (1 + (-1)^n/(12*n)). - _Vaclav Kotesovec_, Mar 02 2014
%t A036256 Table[Sum[Binomial[k, Floor[k/2]], {k, 0, n}], {n, 0, 20}]
%t A036256 Table[Piecewise[{{(1/2)*(-1 - I*Sqrt[3] - (3*Gamma[3 + n]*Hypergeometric2F1Regularized[1, (3 + n)/2, (4 + n)/2, 4])/Gamma[2 + n/2]), Mod[n, 2] == 0}, {((-1 - I*Sqrt[3])*Gamma[(1 + n)/2] - 4*n!*(Hypergeometric2F1Regularized[1, (2 + n)/2, (3 + n)/2, 4] + (2 + n)*Hypergeometric2F1Regularized[1, (4 + n)/2, (5 + n)/2, 4]))/(2*Gamma[(1 + n)/2]), Mod[n, 2] == 1}}], {n, 0, 20}] // Expand
%o A036256 (PARI) for(n=0,50, print1(sum(k=0,n, binomial(k,floor(k/2))), ", ")) \\ _G. C. Greubel_, Jan 24 2017
%Y A036256 Partial sums of A001405.
%Y A036256 Cf. A145972. - _Gary W. Adamson_, Oct 25 2008
%K A036256 nonn
%O A036256 0,2
%A A036256 _N. J. A. Sloane_