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%I #47 Sep 08 2022 08:44:50
%S 1,13,133,1321,13081,129493,1281853,12689041,125608561,1243396573,
%T 12308357173,121840175161,1206093394441,11939093769253,
%U 118184844298093,1169909349211681,11580908647818721,114639177128975533,1134810862641936613,11233469449290390601
%N Numbers k such that 1^5 + 2^5 + ... + k^5 is a square.
%C Partial sums of A004291 or convolution of A040000 with A054320. - _R. J. Mathar_, Oct 26 2009
%C 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
%C 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
%H Colin Barker, <a href="/A031138/b031138.txt">Table of n, a(n) for n = 1..1000</a>
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/HexNumber.html">Hex Number</a>
%H <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (11, -11, 1).
%F a(n) = 11*(a(n-1) - a(n-2)) + a(n-3).
%F a(n) = -1/2 + ((3 - sqrt(6))/4)*(5 + 2*sqrt(6))^n + ((3 + sqrt(6))/4)*(5 - 2*sqrt(6))^n.
%F 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.
%F 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
%F contfrac(sqrt(6)/A054320(n))[4]/2 - _Thomas Baruchel_, Dec 02 2003
%F G.f.: x*(1+x)^2/((1-x)*(x^2-10*x+1)). - _R. J. Mathar_, Oct 26 2009
%e a(2) = 13 because 1^5+2^5+...13^5 = 1001^2; a(1) = 1 because 1^5 = 1^2.
%t LinearRecurrence[{11,-11,1},{1,13,133},20 ] (* _Harvey P. Dale_, Oct 23 2012 *)
%o (PARI) isok(n) = issquare(sum(i=1, n, i^5)); \\ _Michel Marcus_, Dec 28 2013
%o (PARI) Vec(x*(1+x)^2/((1-x)*(x^2-10*x+1)) + O(x^40)) \\ _Colin Barker_, Sep 06 2015
%o (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
%Y Cf. A000539, A006061, A054320, A007667.
%K easy,nonn
%O 1,2
%A _Ignacio Larrosa Cañestro_, entry revised Feb 27 2000
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