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

1, 2, 4, 7, 13, 23, 43, 78, 148, 274, 526, 988, 1912, 3628, 7060, 13495, 26365, 50675, 99295, 191673, 376429, 729145, 1434577, 2786655, 5490811, 10691111, 21091711, 41150011, 81266611, 158825371, 313942891, 614483086, 1215563476

Offset: 0

Author

N. J. A. Sloane

Keywords

• nonn

Comments

Equals row sums of triangle A145972. - Gary W. Adamson, Oct 25 2008

a(n-1) is the graph bandwidth of the n-hypercube graph Q_n. - Eric W. Weisstein, Jul 12 2011

Links

• G. C. Greubel, Table of n, a(n) for n = 0..1000
• L. H. Harper, Optimal numberings and isoperimetric problems on graphs, J. Comb. Th. 1 (1966), 385-393.
• Eric Weisstein's World of Mathematics, Graph Bandwidth
• Eric Weisstein's World of Mathematics, Hypercube Graph

Crossrefs

Partial sums of A001405.

Cf. A145972. - Gary W. Adamson, Oct 25 2008

Programs

Mathematica

Table[Sum[Binomial[k, Floor[k/2]], {k, 0, n}], {n, 0, 20}]
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

PARI

for(n=0,50, print1(sum(k=0,n, binomial(k,floor(k/2))), ", ")) \\ G. C. Greubel, Jan 24 2017

Formula

G.f.: 2/((1-z)*(1-2*z+sqrt(1-4*z^2))). - Emeric Deutsch, Nov 25 2003

a(n) ~ 2^(n+3/2) / sqrt(Pi*n) * (1 + (-1)^n/(12*n)). - Vaclav Kotesovec, Mar 02 2014