A128624 Row sums of A128623.

1, 4, 12, 24, 45, 72, 112, 160, 225, 300, 396, 504, 637, 784, 960, 1152, 1377, 1620, 1900, 2200, 2541, 2904, 3312, 3744, 4225, 4732, 5292, 5880, 6525, 7200, 7936, 8704, 9537, 10404, 11340, 12312, 13357, 14440, 15600, 16800, 18081, 19404, 20812, 22264, 23805

Offset: 1

Author

Gary W. Adamson, Mar 14 2007

Keywords

• nonn
• easy

Comments

Also the number of (w,x,y) with all terms in {0,...,n-1} and w <= R <= x, where R = max(w,x,y)-min(w,x,y), see A212959. - Clark Kimberling, Jun 10 2012

Links

• Colin Barker, 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. A128621, A128623, A212959.

Cf. A094728 (diagonal row sums).

Programs

Magma

[n*((n+1)^2-1+(n mod 2))/4: n in [1..50]]; // G. C. Greubel, Mar 12 2024

Mathematica

Table[n*(n^2 +2*n +Mod[n,2])/4, {n,50}] (* G. C. Greubel, Mar 12 2024 *)

PARI

Vec(x*(1+2*x+3*x^2)/((1-x)^4*(1+x)^2) + O(x^100)) \\ Colin Barker, Jan 31 2016

SageMath

[n*((n+1)^2-1+(n%2))//4 for n in range(1,51)] # G. C. Greubel, Mar 12 2024

Formula

G.f.: x*(1+2*x+3*x^2) / ((1+x)^2*(1-x)^4). - R. J. Mathar, Jun 27 2012

From Colin Barker, Jan 31 2016: (Start)

a(n) = n*(2*n^2 + 4*n + 1 - (-1)^n)/8.

a(n) = n^2*(n + 2)/4 for n even.

a(n) = n*(n^2 + 2*n + 1)/4 for n odd. (End)

From G. C. Greubel, Mar 12 2024: (Start)

a(n) = Sum_{k=0..floor((n-1)/2)} A094728(n, k).

E.g.f.: (1/8)*x*(exp(-x) + (7 + 10*x + 2*x^2)*exp(x)). (End)

Extensions

Incorrect formula removed by R. J. Mathar, Jun 27 2012