login
a(n) = A188491(n-1) + A188495(n-1) + A188497(n-1).
7

%I #23 Jan 06 2016 13:53:33

%S 0,0,2,6,14,31,104,344,1084,3236,9784,29964,92241,282780,865064,

%T 2646292,8102454,24813838,75982346,232630527,712230076,2180675264,

%U 6676819512,20443032008,62591840320,191641545768,586762729889,1796535598952,5500587026592

%N a(n) = A188491(n-1) + A188495(n-1) + A188497(n-1).

%C 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(j) <= 2+j for j = 1,2, and p(j) >= j-2 for j = 4,5.

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

%C This is row 7 of Kløve's Table 3.

%H Nathaniel Johnston, <a href="/A188493/b188493.txt">Table of n, a(n) for n = 0..124</a>

%H Torleiv Kløve, <a href="http://www.ii.uib.no/publikasjoner/texrap/pdf/2008-376.pdf"> Spheres of Permutations under the Infinity Norm - Permutations with limited displacement. </a> Reports in Informatics, Department of Informatics, University of Bergen, Norway, no. 376, November 2008.

%F G.f.: -(x^10+2*x^9+2*x^7 +4*x^6-2*x^5-6*x^4 -9*x^3-2*x^2+2*x+2) *x^2 / (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

%p with (LinearAlgebra):

%p A188493:= n-> `if` (n<=1, 0, Permanent (Matrix (n, (i, j)->

%p `if` (abs(j-i)<4 and [i, j]<>[4, 1] and [i, j]<>[5, 2] and [i, j]<>[1, 4] and [i, j]<>[2, 5], 1, 0)))):

%p seq (A188493(n), n=0..20);

%t a[n_] := Permanent[Table[If[Abs[j - i] < 4 && {i, j} != {4, 1} && {i, j} != {5, 2} && {i, j} != {1, 4} && {i, j} != {2, 5}, 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 *)

%K nonn

%O 0,3

%A _N. J. A. Sloane_, Apr 01 2011

%E Name and comments edited, and a(12)-a(28) from _Nathaniel Johnston_, Apr 08 2011