A002604 - OEIS

%I #31 Sep 08 2022 08:44:31

%S 1,2,65,730,4097,15626,46657,117650,262145,531442,1000001,1771562,

%T 2985985,4826810,7529537,11390626,16777217,24137570,34012225,47045882,

%U 64000001,85766122,113379905,148035890

%N a(n) = n^6 + 1.

%C Because of Fermat's little theorem, a(n) is never divisible by 7. - _Altug Alkan_, Apr 08 2016

%H Vincenzo Librandi, <a href="/A002604/b002604.txt">Table of n, a(n) for n = 0..1000</a>

%H <a href="/index/Rec#order_07">Index entries for linear recurrences with constant coefficients</a>, signature (7,-21,35,-35,21,-7,1).

%F G.f. (-1 + 5*x - 72*x^2 - 282*x^3 - 317*x^4 - 51*x^5 - 2*x^6) / (x - 1)^7. - _R. J. Mathar_, Aug 06 2012

%F Sum_{n>=0} 1/a(n) = 1/2 + Pi * (coth(Pi) + (sinh(Pi) + sqrt(3)*sin(sqrt(3)*Pi)) / (cosh(Pi) - cos(sqrt(3)*Pi))) / 6 = 1.5171007340332164261529... . - _Vaclav Kotesovec_, Feb 14 2015

%F Sum_{n>=0} (-1)^n/a(n) = 1/2 + Pi/(6*sinh(Pi)) + Pi * (sqrt(3)*cosh(Pi/2) * sin((sqrt(3)*Pi)/2) + cos((sqrt(3)*Pi)/2) * sinh(Pi/2)) / (3*(cosh(Pi) - cos(sqrt(3)*Pi))) = 0.514210347292695053493... . - _Vaclav Kotesovec_, Feb 14 2015

%t Table[n^6+1,{n,0,40}] (* _Vladimir Joseph Stephan Orlovsky_, Apr 15 2011 *)

%t LinearRecurrence[{7,-21,35,-35,21,-7,1},{1,2,65,730,4097,15626,46657},30] (* _Harvey P. Dale_, Jul 28 2021 *)

%o (PARI) a(n)=n^6+1

%o (Magma) [n^6 + 1: n in [0..50]]; // _Vincenzo Librandi_, May 02 2011

%Y Equals A001014 + 1. Cf. A024004, A002522.

%K nonn,easy

%O 0,2

%A _N. J. A. Sloane_