A330520 Sum of even integers <= n times the sum of odd integers <= n.

0, 0, 2, 8, 24, 54, 108, 192, 320, 500, 750, 1080, 1512, 2058, 2744, 3584, 4608, 5832, 7290, 9000, 11000, 13310, 15972, 19008, 22464, 26364, 30758, 35672, 41160, 47250, 54000, 61440, 69632, 78608, 88434, 99144, 110808, 123462, 137180, 152000, 168000, 185220, 203742, 223608, 244904, 267674

Offset: 0

Author

J. Stauduhar, Dec 17 2019

Keywords

• nonn
• easy

Comments

Number of crossings in a grid formed by drawing n parallel infinite-length lines perpendicular to the previous number of lines.

The sum of odd integers <= n is m^2 where m = round(n/2) is their number. The sum of even integers <= n is k(k+1) where k = floor(n/2) is their number. So a(n) = m^2*k(k+1), where the factor m appears three times. - M. F. Hasler, Dec 19 2019

Links

• Harvey P. Dale, Table of n, a(n) for n = 0..1000
• Index entries for linear recurrences with constant coefficients, signature (3,-1,-5,5,1,-3,1).

Crossrefs

Cf. A179824, A085537, A007009, A002620.

Cf. A000290 (sum of odd integers), A002378 (sum of even integers).

Programs

Mathematica

CoefficientList[Series[2 (x^2 + x + 1) x^2/((x + 1)^2*(1 - x)^5), {x, 0, 45}], x] (* Michael De Vlieger, Dec 22 2019 *)
LinearRecurrence[{3,-1,-5,5,1,-3,1},{0,0,2,8,24,54,108},50] (* Harvey P. Dale, Dec 29 2021 *)

PARI

apply( A330520(n)=n\2*(n\2+1)*(n\/2)^2, [0..99]) \\ M. F. Hasler, Dec 19 2019

Formula

G.f.: 2*(x^2+x+1)*x^2/((x+1)^2*(1-x)^5).

a(n) = 2 * A007009(n-1) for n>1.

a(2k+i) = (k+i)^3 (k+1-i), with i = 0 or 1. - M. F. Hasler, Dec 19 2019

a(n) = A002378(floor(n/2)) * A000290(ceiling(n/2)). - Bernard Schott, Dec 22 2019