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A093328
a(n) = 2*n^2 + 3.
16
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
OFFSET
0,1
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
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.
FORMULA
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
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 *)
PROG
(PARI) a(n)=2*n^2+3; \\ Zak Seidov, Sep 27 2015
(Magma) [2*n^2+3: n in [0..50]]; // Vincenzo Librandi, Jul 08 2012
CROSSREFS
a(n) = A005893(n)+1 = A058331(n)+2 = A001105(n)+3.
a(n+2) = A154685(n+1,n+2).
Sequence in context: A074710 A057460 A269112 * A045515 A298342 A293317
KEYWORD
nonn,easy
AUTHOR
Ralf Stephan, Apr 25 2004
EXTENSIONS
Simpler definition and new offset from Paul F. Brewbaker, Jun 23 2009
Edited by N. J. A. Sloane, Jun 27 2009
STATUS
approved