A002529 a(n) = A002527(n+1) - A002527(n) - A002526(n).

0, 0, 2, 6, 18, 46, 146, 460, 1436, 4352, 13252, 40532, 124396, 381140, 1166708, 3570684, 10932274, 33475170, 102499334, 313825690, 960844358, 2941873064, 9007393480, 27578681888, 84439657768, 258534813320, 791574775192, 2423623112104, 7420586212184, 22720153701768, 69563959091138

Offset: 0

Author

N. J. A. Sloane

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 and p(j) <= 2+j for j = 1,2.

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, ones on its three subdiagonals (with the exception of zeros in the (4,1) and (5,2)-entries), and is zero elsewhere.

This is row 3 of Kløve's Table 3.

References

• N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
• N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Links

• Alois P. Heinz, 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.
• R. Lagrange, Quelques résultats dans la métrique des permutations, Annales Scientifiques de l'École Normale Supérieure, Paris, 79 (1962), 199-241.
• Index entries for linear recurrences with constant coefficients, signature (2, 2, 0, 10, 8, -2, -16, -10, -2, 4, 2, 0, 2, 1).

Programs

Maple

with (LinearAlgebra):
A002529:= n-> `if` (n<=1, 0, Permanent (Matrix (n, (i, j)->
              `if` (abs(j-i)<4 and [i, j]<>[4, 1] and [i, j]<>[5, 2], 1, 0)))):
seq (A002529(n), n=0..20);

Mathematica

a[n_] := If [n <= 1, 0, Permanent[Table[If[Abs[j-i]<4 && {i, j} != {4, 1} && {i, j} != {5, 2}, 1, 0], {i, 1, n}, {j, 1, n}]]]; Table[a[n], {n, 0, 30}] (* Jean-François Alcover, Mar 12 2014, after Maple *)

Formula

a(n) = A188379(n+1) - A188492(n) - A188493(n). - Nathaniel Johnston, Apr 08 2011

G.f.: 2*x^2 * (x^4+x^3-x^2-x-1) / (x^14+2*x^13+2*x^11 +4*x^10 -2*x^9 -10*x^8 -16*x^7-2*x^6 +8*x^5+10*x^4 +2*x^2+2*x-1). - Alois P. Heinz, Apr 08 2011

Extensions

Name and comments edited and terms a(12)-a(30) from Nathaniel Johnston, Apr 08 2011