%I #64 Feb 26 2024 20:11:20
%S 0,1,2,33,244,1025,3126,7777,16808,32769,59050,100001,161052,248833,
%T 371294,537825,759376,1048577,1419858,1889569,2476100,3200001,4084102,
%U 5153633,6436344,7962625,9765626,11881377,14348908,17210369,20511150,24300001,28629152,33554433
%N a(n) = n^5 + 1.
%H Vincenzo Librandi, <a href="/A002561/b002561.txt">Table of n, a(n) for n = -1..1000</a>
%H <a href="/index/Rec#order_06">Index entries for linear recurrences with constant coefficients</a>, signature (6,-15,20,-15,6,-1).
%F a(n) = A000584(n) + 1, for n >= 0.
%F a(n) = 6*a(n-1) - 15*a(n-2) + 20*a(n-3) - 15*a(n-4) + 6*a(n-5) - a(n-6), n > 4. - _Harvey P. Dale_, Aug 13 2013
%F G.f.: ( 1 - 4*x + 36*x^2 + 56*x^3 + 31*x^4 ) / (x-1)^6. - _R. J. Mathar_, Jul 28 2014
%F E.g.f.: (1 + x + 15*x^2 + 25*x^3 + 10*x^4 + x^5)*exp(x). - _G. C. Greubel_, Oct 24 2018
%F Sum_{n>=1} 1/a(n) = 1/2 + Sum_{n>=1} (-1)^(n+1) * (zeta(5*n) - 1) = 0.5359628431... - _Amiram Eldar_, Nov 06 2020
%t Table[n^5+1,{n,-1,40}] (* _Vladimir Joseph Stephan Orlovsky_, Apr 15 2011 *)
%t LinearRecurrence[{6,-15,20,-15,6,-1},{0,1,2,33,244,1025},40] (* _Harvey P. Dale_, Aug 13 2013 *)
%o (Magma) [n^5+1: n in [-1..30]]; // _Vincenzo Librandi_, Jun 07 2013
%o (PARI) a(n)=n^5+1 \\ _Charles R Greathouse IV_, Oct 07 2015
%o (GAP) List([1..30],n->n^5+1); # _Muniru A Asiru_, Oct 23 2018
%o (Sage) [n^5 + 1 for n in (-1..30)] # _G. C. Greubel_, Nov 20 2018
%Y Cf. A000584.
%K nonn,easy
%O -1,3
%A _N. J. A. Sloane_