A079921 Solution to the Dancing School Problem with n girls and n+2 boys: f(n,2).

3, 7, 14, 26, 46, 79, 133, 221, 364, 596, 972, 1581, 2567, 4163, 6746, 10926, 17690, 28635, 46345, 75001, 121368, 196392, 317784, 514201, 832011, 1346239, 2178278, 3524546, 5702854, 9227431, 14930317, 24157781, 39088132, 63245948, 102334116, 165580101

Offset: 1

Author

Jaap Spies, Jan 28 2003

Keywords

• nonn

Comments

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.

With offset 4, number of 132-avoiding two-stack sortable permutations which contain exactly one subsequence of type 123.

Links

• Harvey P. Dale, Table of n, a(n) for n = 1..1000
• Jaap Spies, Dancing School Problems, Nieuw Archief voor Wiskunde 5/7 nr. 4, Dec 2006, pp. 283-285.
• Jaap Spies, Dancing School Problems, Permanent solutions of Problem 29.
• E. S. Egge and T. Mansour, 132-avoiding two-stack sortable permutations....
• Index entries for linear recurrences with constant coefficients, signature (3,-2,-1,1).

Crossrefs

Cf. A000045, A079908-A079928, A001924.

Cf. Essentially the same as A001924.

Programs

Maple

with(genfunc): Fz := 1/((-1+z)^2 * (1-z-z^2)); seq(rgf_term(Fz,z,n), n=1..30);

Mathematica

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 *)
LinearRecurrence[{3,-2,-1,1},{3,7,14,26},40] (* Harvey P. Dale, Oct 17 2022 *)

Formula

a(n) = a(n-1)+a(n-2)+n+1, a(1)=3, a(2)=7.

G.f.: 1/((1-x)^2*(1-x-x^2)).

F(n+5) - n - 4, F(n) = A000045(n).

a(n) = 3*a(n-1)-2*a(n-2)-a(n-3)+a(n-4). - Wesley Ivan Hurt, Dec 03 2021

Extensions

More terms from Jaap Spies, Dec 15 2006