cp's OEIS Frontend

A002528 a(n) = A188491(n+1) - A188494(n) - A002526(n).

%I A002528 M1256 N0480 #37 Dec 27 2024 22:37:52
%S A002528 0,0,2,4,12,32,108,336,1036,3120,9540,29244,89768,274788,840936,
%T A002528 2573972,7881922,24135000,73897320,226249264,692714696,2120941424,
%U A002528 6493883944,19882820480,60876609464,186390208744,570684661408,1747307671896,5349860697088
%N A002528 a(n) = A188491(n+1) - A188494(n) - A002526(n).
%C A002528 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, and p(2) <= 4.
%C A002528 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 (3,1), (4,1), and (5,2)-entries), and is zero elsewhere.
%C A002528 This is row 9 of Kløve's Table 3.
%D A002528 N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
%D A002528 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%H A002528 Nathaniel Johnston, <a href="/A002528/b002528.txt">Table of n, a(n) for n = 0..90</a>
%H A002528 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 A002528 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 A002528 <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 A002528 a(n) = A002527(n-1) + A188491(n-1). - _Nathaniel Johnston_, Apr 10 2011
%F A002528 G.f.: -2*x^2 / ((x -1)*(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 16 2014
%p A002528 with(LinearAlgebra):
%p A002528 A002528:= n-> `if` (n<=1, 0, Permanent (Matrix (n, (i, j)->
%p A002528               `if` (abs(j-i)<4 and [i, j]<>[3, 1] and [i, j]<>[4, 1] and [i, j]<>[5, 2], 1, 0)))):
%t A002528 a[n_] := Permanent[Table[If[Abs[j - i] < 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 07 2016, adapted from Maple *)
%t A002528 CoefficientList[Series[2 x^2 / ((1 - x) (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)), {x, 0, 33}], x] (* _Vincenzo Librandi_, Jan 07 2016 *)
%t A002528 LinearRecurrence[{2,2,0,10,8,-2,-16,-10,-2,4,2,0,2,1},{0,0,2,4,12,32,108,336,1036,3120,9540,29244,89768,274788},20] (* _Harvey P. Dale_, Jan 04 2020 *)
%o A002528 (PARI) concat([0,0], Vec(-2*x^2 / ((x -1)*(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 16 2014
%K A002528 nonn,easy
%O A002528 0,3
%A A002528 _N. J. A. Sloane_
%E A002528 Name and comments edited, and a(12)-a(28) from _Nathaniel Johnston_, Apr 10 2011