A159469 Maximum remainder when (k + 1)^n + (k - 1)^n is divided by k^2 for variable n and k > 2.

6, 8, 20, 24, 42, 48, 72, 80, 110, 120, 156, 168, 210, 224, 272, 288, 342, 360, 420, 440, 506, 528, 600, 624, 702, 728, 812, 840, 930, 960, 1056, 1088, 1190, 1224, 1332, 1368, 1482, 1520, 1640, 1680, 1806, 1848, 1980, 2024, 2162, 2208, 2352, 2400, 2550, 2600

Offset: 3

Author

Gaurav Kumar, Apr 13 2009

Keywords

• nonn
• easy

Examples

			For n = 3, maxr => 3*3 - 3 = 6 since 3 is odd.
For n = 4, maxr => 4*4 - 2*4 = 8 since 4 is even.
		

Links

• Iain Fox, Table of n, a(n) for n = 3..10000
• Project Euler, Problem 120: Square remainders
• Iain Fox, Proof for recursive definition and generating function
• Index entries for linear recurrences with constant coefficients, signature (1,2,-2,-1,1).

Crossrefs

Cf. A050187.

Programs

Mathematica

LinearRecurrence[{1,2,-2,-1,1},{6,8,20,24,42},50] (* Harvey P. Dale, Apr 18 2018 *)

PARI

a(n) = if (n % 2, n^2 - n, n^2 - 2*n); \\ Michel Marcus, Aug 26 2013

PARI

first(n) = Vec(x^3*(-6-2*x)/((x+1)^2*(x-1)^3) + O(x^(n+3))) \\ Iain Fox, Nov 26 2017

Formula

maxr(n) = n*n - 2*n if n is even, and n*n - n if n is odd.

G.f.: x^3*(-6-2*x)/((x+1)^2*(x-1)^3). - Maksym Voznyy (voznyy(AT)mail.ru), Jul 26 2009 (proved by Iain Fox, Nov 26 2017)

a(n) = 2*A050187(n). - R. J. Mathar, Aug 08 2009 (proved by Iain Fox, Nov 27 2017)

a(n) = a(n-1) + 2*a(n-2) - 2*a(n-3) - a(n-4) + a(n-5) for n > 7. - Colin Barker, Oct 29 2017 (proved by Iain Fox, Nov 26 2017)

a(n) = n^2 - n*(3 + (-1)^n)/2. - Iain Fox, Nov 26 2017

From Iain Fox, Nov 27 2017: (Start)

a(n) = A000290(n) - A022998(n).

a(n) = 2*A093005(n-2) + A168273(n-1).

a(n) = (4*(A152749(n-2)) + A091574(n-1) - A010719(n-1))/3.

E.g.f.: x*(exp(x)*x - sinh(x)).

(End)