%I A031138
%S A031138 1,13,133,1321,13081,129493,1281853,12689041,125608561,1243396573,
%T A031138 12308357173,121840175161,1206093394441,11939093769253,118184844298093
%N A031138 1^5+2^5+...+n^5 is a square.
%C A031138 Partial sums of A004291 or convolution of A040000 with A054320. [From 
               R. J. Mathar (mathar(AT)strw.leidenuniv.nl), Oct 26 2009]
%H A031138 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/
               HexNumber.html">Hex Number</a>
%F A031138 a(n) =11*(a(n-1)-a(n-2)) + a(n-3); a(n)=-1/2+((3-sqrt(6))/4)*(5+2sqrt(6))^n+((3+sqrt(6))/
               4)*(5-2sqrt(6))^n.
%F A031138 a(n)^2+(a(n)+1)^2=(b(n)-1)^2+b(n)^2+(b(n)+1)^2=c(n)=3d(n)+2; where b(n) 
               is A054320, c(n) is A007667 and d(n) is A006061
%F A031138 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 (blekraj(AT)yahoo.com), Jul 08 2002
%F A031138 contfrac(sqrt(6)/A054320(n))[4]/2 - Thomas Baruchel (baruchel(AT)users.sourceforge.net), 
               Dec 02 2003
%F A031138 G.f.: x*(1+x)^2/)(1-x)*(x^2-10*x+1)). [From R. J. Mathar (mathar(AT)strw.leidenuniv.nl), 
               Oct 26 2009]
%e A031138 a(2)=13 because 1^5+2^5+...13^5=1001^2; a(1)=1 because 1^5=1^2
%Y A031138 Cf. A006061, A054320, A007667.
%Y A031138 Sequence in context: A037617 A081042 A016153 this_sequence A097166 A073556 
               A154999
%Y A031138 Adjacent sequences: A031135 A031136 A031137 this_sequence A031139 A031140 
               A031141
%K A031138 easy,nonn
%O A031138 1,2
%A A031138 Ignacio Larrosa Canestro (ignacio.larrosa(AT)eresmas.net) entry revised 
               Feb 27 2000