A190798 Maximum value of k^2 * (n-k).

0, 0, 1, 4, 9, 18, 32, 50, 75, 108, 147, 196, 256, 324, 405, 500, 605, 726, 864, 1014, 1183, 1372, 1575, 1800, 2048, 2312, 2601, 2916, 3249, 3610, 4000, 4410, 4851, 5324, 5819, 6348, 6912, 7500, 8125, 8788, 9477, 10206, 10976, 11774, 12615, 13500, 14415, 15376, 16384, 17424, 18513, 19652, 20825, 22050, 23328, 24642, 26011, 27436, 28899, 30420, 32000

Offset: 0

Author

Franklin T. Adams-Watters, May 20 2011

Keywords

• nonn
• easy

Links

• G. C. Greubel, Table of n, a(n) for n = 0..5000
• Index entries for linear recurrences with constant coefficients, signature (2,-1,2,-4,2,-1,2,-1).

Crossrefs

Cf. A002620 (max of k * (n-k)).

Programs

Magma

m:=50; R:=PowerSeriesRing(Integers(), m); [0,0] cat Coefficients(R!(x^2*(1+x)^2*(1+x^2)/((1-x)^4*(1+x+x^2)^2))); // G. C. Greubel, Aug 13 2018

Mathematica

CoefficientList[Series[x^2*(1+x)^2*(1+x^2)/((1-x)^4*(1+x+x^2)^2), {x, 0, 50}], x] (* G. C. Greubel, Aug 13 2018 *)

PARI

a(n)=my(k=2*n\/3);k^2*(n-k) \\ Charles R Greathouse IV, May 20 2011

Formula

a(n) = k^2 * (n-k), where k = round(2*n/3).

a(3*n) = 4*n^3, a(3*n-1) = n*(2*n-1)^2, a(3*n+1) = n*(2*n+1)^2.

O.g.f.: x^2*(1+x)^2*(1+x^2)/((1-x)^4*(1+x+x^2)^2).

a(n) = 2*a(n-1) - a(n-2) + 2*a(n-3) - 4*a(n-4) + 2*a(n-5) - a(n-6) + 2*a(n-7) - a(n-8) for n >= 8.

a(-n) = -a(n). - Michael Somos, May 22 2011