A031138 Numbers k such that 1^5 + 2^5 + ... + k^5 is a square.

1, 13, 133, 1321, 13081, 129493, 1281853, 12689041, 125608561, 1243396573, 12308357173, 121840175161, 1206093394441, 11939093769253, 118184844298093, 1169909349211681, 11580908647818721, 114639177128975533, 1134810862641936613, 11233469449290390601

Offset: 1

Author

Ignacio Larrosa Cañestro, entry revised Feb 27 2000

Keywords

• easy
• nonn

Comments

Partial sums of A004291 or convolution of A040000 with A054320. - R. J. Mathar, Oct 26 2009

This is a 6th-degree Diophantine equation 12*m^2 = n^2*(n+1)^2*(2*n^2 + 2*n - 1) which reduces to the generalized Pell equation 6*q^2 = (2*n + 1)^2 - 3 where q = 3*m/(n*(n+1)), so there is no surprise that the solutions satisfy a linear recurrent equation. - Charles R Greathouse IV, Max Alekseyev, Oct 22 2012

Also k such that k^2 + (k+1)^2 is equal to the sum of three consecutive squares, for example 13^2 + 14^2 = 10^2 + 11^2 + 12^2. - Colin Barker, Sep 06 2015

Examples

			a(2) = 13 because 1^5+2^5+...13^5 = 1001^2; a(1) = 1 because 1^5 = 1^2.
		

Links

• Colin Barker, Table of n, a(n) for n = 1..1000
• Eric Weisstein's World of Mathematics, Hex Number
• Index entries for linear recurrences with constant coefficients, signature (11, -11, 1).

Crossrefs

Cf. A000539, A006061, A054320, A007667.

Programs

Magma

[Round(-1/2 + ((3 - Sqrt(6))/4)*(5 + 2*Sqrt(6))^n + ((3 + Sqrt(6) )/4)*(5 - 2*Sqrt(6))^n): n in [0..50]]; // G. C. Greubel, Nov 04 2017

Mathematica

LinearRecurrence[{11,-11,1},{1,13,133},20 ] (* Harvey P. Dale, Oct 23 2012 *)

PARI

isok(n) = issquare(sum(i=1, n, i^5)); \\ Michel Marcus, Dec 28 2013

PARI

Vec(x*(1+x)^2/((1-x)*(x^2-10*x+1)) + O(x^40)) \\ Colin Barker, Sep 06 2015

Formula

a(n) = 11*(a(n-1) - a(n-2)) + a(n-3).

a(n) = -1/2 + ((3 - sqrt(6))/4)*(5 + 2*sqrt(6))^n + ((3 + sqrt(6))/4)*(5 - 2*sqrt(6))^n.

a(n)^2 + (a(n) + 1)^2 = (b(n) - 1)^2 + b(n)^2 + (b(n) + 1)^2 = c(n) = 3*d(n) + 2; where b(n) is A054320, c(n) is A007667 and d(n) is A006061.

a(n) = 10*a(n-1) - a(n-2) + 4; a(0) = a(1) = 1. Also sum of first a(n) fifth powers is a square m^2, where m has factors A000217{a(n)} and A054320(n). - Lekraj Beedassy, Jul 08 2002

contfrac(sqrt(6)/A054320(n))[4]/2 - Thomas Baruchel, Dec 02 2003

G.f.: x*(1+x)^2/((1-x)*(x^2-10*x+1)). - R. J. Mathar, Oct 26 2009