A093328 a(n) = 2*n^2 + 3.

3, 5, 11, 21, 35, 53, 75, 101, 131, 165, 203, 245, 291, 341, 395, 453, 515, 581, 651, 725, 803, 885, 971, 1061, 1155, 1253, 1355, 1461, 1571, 1685, 1803, 1925, 2051, 2181, 2315, 2453, 2595, 2741, 2891, 3045, 3203, 3365, 3531, 3701, 3875, 4053, 4235, 4421, 4611

Offset: 0

Author

Ralf Stephan, Apr 25 2004

Keywords

• nonn
• easy

Comments

Number of 132-avoiding two-stack sortable permutations which also avoid 4321.

Conjecture: no perfect powers. - Zak Seidov, Sep 27 2015

Numbers k such that 2*k - 6 is a square. - Bruno Berselli, Nov 08 2017

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..1000
• Steven Edwards and William Griffiths, Generalizations of Delannoy and cross polytope numbers, Fib. Q., Vol. 55, No. 4 (2017), pp. 356-366.
• Steven Edwards and William Griffiths, On Generalized Delannoy Numbers, J. Int. Seq., Vol. 23 (2020), Article 20.3.6.
• Eric S. Egge and Toufik Mansour, 132-avoiding two-stack sortable permutations, Fibonacci numbers, and Pell numbers, Discrete Applied Mathematics, Vol. 143, No. 1-3 (2004), pp. 72-83; arXiv preprint, arXiv:math/0205206 [math.CO], 2002.
• Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Crossrefs

Cf. A001105, A005893, A058331, A154685.

Programs

Magma

[2*n^2+3: n in [0..50]]; // Vincenzo Librandi, Jul 08 2012

Mathematica

Table[2 n^2 + 3, {n, 0, 100}] (* Vladimir Joseph Stephan Orlovsky, Feb 15 2011*)
CoefficientList[Series[(3 - 4 x + 5 x^2)/(1 - x)^3, {x, 0, 50}], x] (* Vincenzo Librandi, Jul 08 2012 *)
LinearRecurrence[{3, -3, 1}, {3, 5, 11}, 50] (* Harvey P. Dale, Apr 03 2016 *)

PARI

a(n)=2*n^2+3; \\ Zak Seidov, Sep 27 2015

Formula

a(n) = A005893(n)+1 = A058331(n)+2 = A001105(n)+3.

a(n+2) = A154685(n+1,n+2).

From Vincenzo Librandi, Jul 08 2012: (Start)

G.f.: (3 - 4*x + 5*x^2)/(1 - x)^3.

a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3). (End)

Sum_{n>=0} 1/a(n) = (1 + sqrt(3/2)*Pi*coth(sqrt(3/2)*Pi))/6. - Amiram Eldar, Nov 25 2020

E.g.f.: exp(x)*(3 + 2*x + 2*x^2). - Elmo R. Oliveira, Jan 17 2025

Extensions

Simpler definition and new offset from Paul F. Brewbaker, Jun 23 2009

Edited by N. J. A. Sloane, Jun 27 2009