A049778 a(n) = Sum_{k=1..floor((n+1)/2)} T(n,2k-1), array T as in A049777.

1, 3, 9, 17, 32, 50, 78, 110, 155, 205, 271, 343, 434, 532, 652, 780, 933, 1095, 1285, 1485, 1716, 1958, 2234, 2522, 2847, 3185, 3563, 3955, 4390, 4840, 5336, 5848, 6409, 6987, 7617, 8265, 8968, 9690, 10470, 11270, 12131

Offset: 1

Author

Clark Kimberling

Keywords

• nonn

Comments

Principal diagonal of the convolution array A213849. - Clark Kimberling, Jul 04 2012

Links

• G. C. Greubel, Table of n, a(n) for n = 1..1000
• Index entries for linear recurrences with constant coefficients, signature (2,1,-4,1,2,-1).

Crossrefs

Cf. A023855, A049777, A095800, A213849.

Programs

GAP

List([1..50], n-> (3 +10*n +18*n^2 +8*n^3 -3*(-1)^n*(1+2*n))/48); # G. C. Greubel, Dec 12 2019

Magma

[(3 +10*n +18*n^2 +8*n^3 -3*(-1)^n*(1+2*n))/48: n in [1..50]]; // G. C. Greubel, Dec 12 2019

Maple

seq( (3 +10*n +18*n^2 +8*n^3 -3*(-1)^n*(1+2*n))/48, n=1..50); # G. C. Greubel, Dec 12 2019

Mathematica

Table[Floor[(n+1)/2]*(3*(n-1)*(n+2) -(1+Floor[(n+1)/2])*(4*Floor[(n+1)/2]-7))/6, {n,50}] (* G. C. Greubel, Dec 12 2019 *)

PARI

vector(50, n, (3 +10*n +18*n^2 +8*n^3 -3*(-1)^n*(1+2*n))/48) \\ G. C. Greubel, Dec 12 2019

Sage

[(3 +10*n +18*n^2 +8*n^3 -3*(-1)^n*(1+2*n))/48 for n in (1..50)] # G. C. Greubel, Dec 12 2019

Formula

G.f.: x*(1 + x + 2*x^2)/((1-x)^4*(1+x)^2). Pairwise sums of A023855. - Ralf Stephan, May 06 2004

a(n) = Sum_{k=1..n} k*ceiling(k/2). - Vladeta Jovovic, Apr 29 2006

Row sums of triangle A095800^2. - Gary W. Adamson, Dec 12 2007

a(n) = (3 + 10*n + 18*n^2 + 8*n^3 - 3*(-1)^n*(1 + 2*n))/48. - R. J. Mathar, Mar 03 2011

From G. C. Greubel, Dec 12 2019: (Start)

a(n) = m*(3*(n-1)*(n+2) - (m+1)*(4*m-7))/6, where m = floor((n+1)/2).

E.g.f.: ( (3+36*x+42*x^2+8*x^3)*exp(x) - 3*(1-2*x)*exp(-x) )/48. (End)