A005491 - OEIS

%I M3855 #39 Dec 01 2022 17:46:08

%S 1,5,15,37,77,141,235,365,537,757,1031,1365,1765,2237,2787,3421,4145,

%T 4965,5887,6917,8061,9325,10715,12237,13897,15701,17655,19765,22037,

%U 24477,27091,29885,32865,36037,39407,42981,46765,50765,54987,59437,64121,69045

%N a(n) = n^3 + 3*n + 1.

%D N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

%H Ivan Panchenko, <a href="/A005491/b005491.txt">Table of n, a(n) for n = 0..1000</a>

%H Simon Plouffe, <a href="https://arxiv.org/abs/0911.4975">Approximations de séries génératrices et quelques conjectures</a>, Dissertation, Université du Québec à Montréal, 1992; arXiv:0911.4975 [math.NT], 2009.

%H Simon Plouffe, <a href="/A000051/a000051_2.pdf">1031 Generating Functions</a>, Appendix to Thesis, Montreal, 1992

%H Earl Glen Whitehead Jr., <a href="http://dx.doi.org/10.1016/0097-3165(78)90061-4">Stirling number identities from chromatic polynomials</a>, J. Combin. Theory, A 24 (1978), 314-317.

%H <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (4,-6,4,-1).

%F a(0)=1, a(1)=5, a(2)=15, a(3)=37, a(n)=4*a(n-1)-6*a(n-2)+4*a(n-3)-a(n-4). - _Harvey P. Dale_, Oct 01 2014

%F From _G. C. Greubel_, Dec 01 2022: (Start)

%F E.g.f.: (1 + 4*x + 3*x^2 + x^3)*exp(x).

%F a(n) = A000578(n) + A016777(n) = A001093(n) + A008585(n). (End)

%p A005491:=(1+z+z**2+3*z**3)/(z-1)**4; # [Conjectured by _Simon Plouffe_ in his 1992 dissertation.]

%t Table[n^3 + 3 n + 1, {n, 0, 50}] (* or *) LinearRecurrence[{4,-6,4,-1},{1,5,15,37},50] (* _Harvey P. Dale_, Oct 01 2014 *)

%o (PARI) a(n)=n^3+3*n+1 \\ _Charles R Greathouse IV_, Oct 07 2015

%o (Magma) [n^3+3*n+1: n in [0..50]]; // _G. C. Greubel_, Dec 01 2022

%o (SageMath) [(n+1)^3 -3*n^2 for n in range(51)] # _G. C. Greubel_, Dec 01 2022

%Y Cf. A000578, A001093, A008585, A016777, A061989.

%K nonn,easy

%O 0,2

%A _N. J. A. Sloane_, _Simon Plouffe_

%E More terms from _Harvey P. Dale_, Oct 01 2014