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%I #26 Oct 17 2022 10:55:25
%S 3,7,14,26,46,79,133,221,364,596,972,1581,2567,4163,6746,10926,17690,
%T 28635,46345,75001,121368,196392,317784,514201,832011,1346239,2178278,
%U 3524546,5702854,9227431,14930317,24157781,39088132,63245948,102334116,165580101
%N Solution to the Dancing School Problem with n girls and n+2 boys: f(n,2).
%C f(g,h) = per(B), the permanent of the (0,1)-matrix B of size g X g+h with b(i,j)=1 if and only if i <= j <= i+h. See A079908 for more information.
%C With offset 4, number of 132-avoiding two-stack sortable permutations which contain exactly one subsequence of type 123.
%H Harvey P. Dale, <a href="/A079921/b079921.txt">Table of n, a(n) for n = 1..1000</a>
%H Jaap Spies, <a href="http://www.nieuwarchief.nl/serie5/pdf/naw5-2006-07-4-283.pdf">Dancing School Problems</a>, Nieuw Archief voor Wiskunde 5/7 nr. 4, Dec 2006, pp. 283-285.
%H Jaap Spies, <a href="http://www.jaapspies.nl/mathfiles/dancingschool.pdf">Dancing School Problems, Permanent solutions of Problem 29</a>.
%H E. S. Egge and T. Mansour, <a href="http://arXiv.org/abs/math.CO/0205206">132-avoiding two-stack sortable permutations...</a>.
%H <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (3,-2,-1,1).
%F a(n) = a(n-1)+a(n-2)+n+1, a(1)=3, a(2)=7.
%F G.f.: 1/((1-x)^2*(1-x-x^2)).
%F F(n+5) - n - 4, F(n) = A000045(n).
%F a(n) = 3*a(n-1)-2*a(n-2)-a(n-3)+a(n-4). - _Wesley Ivan Hurt_, Dec 03 2021
%p with(genfunc): Fz := 1/((-1+z)^2 * (1-z-z^2)); seq(rgf_term(Fz,z,n), n=1..30);
%t CoefficientList[Series[(-z^3 + z^2 + 2*z - 3)/((z - 1)^2 (z^2 + z - 1)), {z, 0, 100}], z] (* _Vladimir Joseph Stephan Orlovsky_, Jul 08 2011 *)
%t LinearRecurrence[{3,-2,-1,1},{3,7,14,26},40] (* _Harvey P. Dale_, Oct 17 2022 *)
%Y Cf. A000045, A079908-A079928, A001924.
%Y Cf. Essentially the same as A001924.
%K nonn
%O 1,1
%A _Jaap Spies_, Jan 28 2003
%E More terms from _Jaap Spies_, Dec 15 2006
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