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

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

Author

N. J. A. Sloane, Simon Plouffe

Keywords

• nonn
• easy

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).

Crossrefs

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

Programs

Magma

[n^3+3*n+1: n in [0..50]]; // G. C. Greubel, Dec 01 2022

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 *)

PARI

a(n)=n^3+3*n+1 \\ Charles R Greathouse IV, Oct 07 2015

SageMath

[(n+1)^3 -3*n^2 for n in range(51)] # G. C. Greubel, Dec 01 2022

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)

Extensions

More terms from Harvey P. Dale, Oct 01 2014