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

0,2

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.

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

Mathematica

Table[Sum[Binomial[k, Floor[k/2]], {k, 0, n}], {n, 0, 20}]
 (* Alternative: *)
Table[3/2 Binomial[2 Floor[n/2], Floor[n/2]] Hypergeometric2F1[1, -Floor[n/2], 1/2 - Floor[n/2], 1/4] + (n - 2 Floor[n/2]) Binomial[2 Floor[n/2] + 1, Floor[n/2]] - 1/2, {n, 0, 20}] (* Eric W. Weisstein, Jul 13 2026 *)
 (* Alternative: *)
CoefficientList[Series[2/((1 - z) (1 - 2 z + Sqrt[1 - 4 z^2])), {z, 0, 20}], z] (* Eric W. Weisstein, Jul 13 2026 *)

Prog

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

Crossrefs

Partial sums of A001405.

Cf. A145972.

Sequence in context: A280027 A026724 A054163 * A093629 A174566 A386399

Adjacent sequences: A036253 A036254 A036255 * A036257 A036258 A036259

Keyword

nonn,changed

Author

N. J. A. Sloane

Status

approved