A188496 a(n) = A188492(n+1) - A188495(n) - A002527(n).

0, 0, 2, 4, 10, 28, 96, 304, 928, 2784, 8504, 26124, 80228, 245544, 751168, 2299184, 7040986, 21561028, 66015398, 202114264, 618817376, 1894692160, 5801169248, 17761879056, 54382725520, 166507388264, 509808051944, 1560917463152, 4779176035680

Offset: 0

Author

N. J. A. Sloane, Apr 01 2011

Keywords

• nonn
• easy

Comments

For n >= 2, a(n) is the number of permutations p on the set [n] with the properties that abs(p(i)-i) <= 3 for all i, p(1) <= 2, p(2) <= 4, and p(4) >= 2.

For n >= 2, a(n) is also the permanent of the n X n matrix that has ones on its diagonal, ones on its three superdiagonals (with the exception of a zero in the (1,4)-entry), ones on its three subdiagonals (with the exception of zeros in the (3,1), (4,1), and (5,2)-entries), and is zero elsewhere.

This is row 11 of Kløve's Table 3.

Links

• Colin Barker, Table of n, a(n) for n = 0..1000
• Torleiv Kløve, Spheres of Permutations under the Infinity Norm - Permutations with limited displacement, Reports in Informatics, Department of Informatics, University of Bergen, Norway, no. 376, November 2008.
• Index entries for linear recurrences with constant coefficients, signature (1,3,3,13,21,19,3,-7,-9,-5,-3,-3,-1).

Programs

Maple

with(LinearAlgebra):
A188496:= n-> `if`(n<=1, 0, Permanent(Matrix(n, (i, j)->
              `if`(abs(j-i)<4 and [i, j]<>[1, 4] and [i, j]<>[3, 1] and [i, j]<>[4, 1] and [i, j]<>[5, 2], 1, 0)))):
seq(A188496(n), n=0..20);

Mathematica

a[n_] := Permanent[Table[If[Abs[j - i] < 4 && {i, j} != {1, 4} && {i, j} != {3, 1} && {i, j} != {4, 1} && {i, j} != {5, 2}, 1, 0], {i, 1, n}, {j, 1, n}] ]; a[1] = 0; Table[an = a[n]; Print["a(", n, ") = ", an]; an, {n, 0, 20}] (* Jean-François Alcover, Jan 06 2016, adapted from Maple *)
LinearRecurrence[{1,3,3,13,21,19,3,-7,-9,-5,-3,-3,-1},{0,0,2,4,10,28,96,304,928,2784,8504,26124,80228},30] (* Harvey P. Dale, Aug 31 2016 *)

PARI

concat([0,0], Vec(x^2*(2*x +2)/(x^13 +3*x^12 +3*x^11 +5*x^10 +9*x^9 +7*x^8 -3*x^7 -19*x^6 -21*x^5 -13*x^4 -3*x^3 -3*x^2 -x +1)  + O(x^100))) \\ Colin Barker, Dec 13 2014

Formula

a(n) = A002527(n-1) + A188495(n-1). - Nathaniel Johnston, Apr 11 2011

G.f.: x^2*(2*x +2)/(x^13 +3*x^12 +3*x^11 +5*x^10 +9*x^9 +7*x^8 -3*x^7 -19*x^6 -21*x^5 -13*x^4 -3*x^3 -3*x^2 -x +1). - Colin Barker, Dec 13 2014

Extensions

Name and comments edited, and a(12)-a(28) from Nathaniel Johnston, Apr 11 2011