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%I #62 Mar 10 2023 02:21:30
%S 3,5,11,21,35,53,75,101,131,165,203,245,291,341,395,453,515,581,651,
%T 725,803,885,971,1061,1155,1253,1355,1461,1571,1685,1803,1925,2051,
%U 2181,2315,2453,2595,2741,2891,3045,3203,3365,3531,3701,3875,4053,4235
%N a(n) = 2*n^2 + 3.
%C Number of 132-avoiding two-stack sortable permutations which also avoid 4321.
%C Conjecture: no perfect powers. - _Zak Seidov_, Sep 27 2015
%C Numbers k such that 2*k - 6 is a square. - _Bruno Berselli_, Nov 08 2017
%H Vincenzo Librandi, <a href="/A093328/b093328.txt">Table of n, a(n) for n = 0..1000</a>
%H Steven Edwards and William Griffiths, <a href="https://www.fq.math.ca/Papers1/55-4/EdwardsGriffiths82617.pdf">Generalizations of Delannoy and cross polytope numbers</a>, Fib. Q., Vol. 55, No. 4 (2017), pp. 356-366.
%H Steven Edwards and William Griffiths, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL23/Griffiths/griffiths51.html">On Generalized Delannoy Numbers</a>, J. Int. Seq., Vol. 23 (2020), Article 20.3.6.
%H Eric S. Egge and Toufik Mansour, <a href="https://doi.org/10.1016/j.dam.2003.12.007">132-avoiding two-stack sortable permutations, Fibonacci numbers, and Pell numbers</a>, Discrete Applied Mathematics, Vol. 143, No. 1-3 (2004), pp. 72-83; <a href="https://arxiv.org/abs/math/0205206">arXiv preprint</a>, arXiv:math/0205206 [math.CO], 2002.
%H <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (3,-3,1).
%F From _Vincenzo Librandi_, Jul 08 2012: (Start)
%F G.f.: (3 - 4*x + 5*x^2)/(1 - x)^3.
%F a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3). (End)
%F Sum_{n>=0} 1/a(n) = (1 + sqrt(3/2)*Pi*coth(sqrt(3/2)*Pi))/6. - _Amiram Eldar_, Nov 25 2020
%t Table[2 n^2 + 3, {n, 0, 100}] (* _Vladimir Joseph Stephan Orlovsky_, Feb 15 2011*)
%t CoefficientList[Series[(3 - 4 x + 5 x^2)/(1 - x)^3, {x, 0, 50}], x] (* _Vincenzo Librandi_, Jul 08 2012 *)
%t LinearRecurrence[{3, -3, 1}, {3, 5, 11}, 50] (* _Harvey P. Dale_, Apr 03 2016 *)
%o (PARI) a(n)=2*n^2+3; \\ _Zak Seidov_, Sep 27 2015
%o (Magma) [2*n^2+3: n in [0..50]]; // _Vincenzo Librandi_, Jul 08 2012
%Y a(n) = A005893(n)+1 = A058331(n)+2 = A001105(n)+3.
%Y a(n+2) = A154685(n+1,n+2).
%K nonn,easy
%O 0,1
%A _Ralf Stephan_, Apr 25 2004
%E Simpler definition and new offset from Paul F. Brewbaker, Jun 23 2009
%E Edited by _N. J. A. Sloane_, Jun 27 2009
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