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A188493 a(n) = A188491(n-1) + A188495(n-1) + A188497(n-1).

%I A188493 #23 Jan 06 2016 13:53:33
%S A188493 0,0,2,6,14,31,104,344,1084,3236,9784,29964,92241,282780,865064,
%T A188493 2646292,8102454,24813838,75982346,232630527,712230076,2180675264,
%U A188493 6676819512,20443032008,62591840320,191641545768,586762729889,1796535598952,5500587026592
%N A188493 a(n) = A188491(n-1) + A188495(n-1) + A188497(n-1).
%C A188493 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 A188493 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 A188493 This is row 7 of Kløve's Table 3.
%H A188493 Nathaniel Johnston, <a href="/A188493/b188493.txt">Table of n, a(n) for n = 0..124</a>
%H A188493 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 A188493 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 A188493 with (LinearAlgebra):
%p A188493 A188493:= n-> `if` (n<=1, 0, Permanent (Matrix (n, (i, j)->
%p A188493               `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 A188493 seq (A188493(n), n=0..20);
%t A188493 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 A188493 nonn
%O A188493 0,3
%A A188493 _N. J. A. Sloane_, Apr 01 2011
%E A188493 Name and comments edited, and a(12)-a(28) from _Nathaniel Johnston_, Apr 08 2011