A005491 a(n) = n^3 + 3*n + 1. (Formerly M3855)

1, 5, 15, 37, 77, 141, 235, 365, 537, 757, 1031, 1365, 1765, 2237, 2787, 3421, 4145, 4965, 5887, 6917, 8061, 9325, 10715, 12237, 13897, 15701, 17655, 19765, 22037, 24477, 27091, 29885, 32865, 36037, 39407, 42981, 46765, 50765, 54987, 59437, 64121, 69045

Offset

0,2

References

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

Links

Ivan Panchenko, Table of n, a(n) for n = 0..1000

Simon Plouffe, Approximations de séries génératrices et quelques conjectures, Dissertation, Université du Québec à Montréal, 1992; arXiv:0911.4975 [math.NT], 2009.

Simon Plouffe, 1031 Generating Functions, Appendix to Thesis, Montreal, 1992

Earl Glen Whitehead Jr., Stirling number identities from chromatic polynomials, J. Combin. Theory, A 24 (1978), 314-317.

Index entries for linear recurrences with constant coefficients, signature (4,-6,4,-1).

Formula

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

From G. C. Greubel, Dec 01 2022: (Start)

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

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

Maple

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

Mathematica

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 *)

Prog

(PARI) a(n)=n^3+3*n+1 \\ Charles R Greathouse IV, Oct 07 2015
(Magma) [n^3+3*n+1: n in [0..50]]; // G. C. Greubel, Dec 01 2022
(SageMath) [(n+1)^3 -3*n^2 for n in range(51)] # G. C. Greubel, Dec 01 2022

Crossrefs

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

Sequence in context: A109818 A146797 A213487 * A348780 A142964 A050488

Adjacent sequences: A005488 A005489 A005490 * A005492 A005493 A005494

Keyword

nonn,easy,changed

Author

N. J. A. Sloane, Simon Plouffe

Extensions

More terms from Harvey P. Dale, Oct 01 2014

Status

approved