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

1,2

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).

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)

Mathematica

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

Prog

(PARI) Vec(x*(1+2*x+3*x^2)/((1-x)^4*(1+x)^2) + O(x^100)) \\ Colin Barker, Jan 31 2016
(Magma) [n*((n+1)^2-1+(n mod 2))/4: n in [1..50]]; // G. C. Greubel, Mar 12 2024
(SageMath) [n*((n+1)^2-1+(n%2))//4 for n in range(1, 51)] # G. C. Greubel, Mar 12 2024

Crossrefs

Cf. A128621, A128623, A212959.

Cf. A094728 (diagonal row sums).

Sequence in context: A008195 A001209 A227587 * A372435 A321879 A328225

Adjacent sequences: A128621 A128622 A128623 * A128625 A128626 A128627

Keyword

nonn,easy

Author

Gary W. Adamson, Mar 14 2007

Extensions

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

Status

approved