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%I M1620 N0633 #38 Jun 25 2023 15:49:07
%S 0,0,2,6,18,46,146,460,1436,4352,13252,40532,124396,381140,1166708,
%T 3570684,10932274,33475170,102499334,313825690,960844358,2941873064,
%U 9007393480,27578681888,84439657768,258534813320,791574775192,2423623112104,7420586212184,22720153701768,69563959091138
%N a(n) = A002527(n+1) - A002527(n) - A002526(n).
%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 and p(j) <= 2+j for j = 1,2.
%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, 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 3 of Kløve's Table 3.
%D N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
%D N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%H Alois P. Heinz, <a href="/A002529/b002529.txt">Table of n, a(n) for n = 0..1000</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.
%H R. Lagrange, <a href="http://archive.numdam.org/article/ASENS_1962_3_79_3_199_0.pdf">Quelques résultats dans la métrique des permutations</a>, Annales Scientifiques de l'École Normale Supérieure, Paris, 79 (1962), 199-241.
%H <a href="/index/Rec#order_14">Index entries for linear recurrences with constant coefficients</a>, signature (2, 2, 0, 10, 8, -2, -16, -10, -2, 4, 2, 0, 2, 1).
%F a(n) = A188379(n+1) - A188492(n) - A188493(n). - _Nathaniel Johnston_, Apr 08 2011
%F 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
%p with (LinearAlgebra):
%p A002529:= 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], 1, 0)))):
%p seq (A002529(n), n=0..20);
%t 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 *)
%K nonn
%O 0,3
%A _N. J. A. Sloane_
%E Name and comments edited and terms a(12)-a(30) from _Nathaniel Johnston_, Apr 08 2011
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