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A163435 Number of different fixed (possibly) disconnected pentominoes bounded tightly by an n X n square. 3

%I

%S 0,0,102,1792,11550,46848,144550,371712,838782,1715200,3247398,

%T 5779200,9774622,15843072,24766950,37531648,55357950,79736832,

%U 112466662,155692800,211949598,284204800,375906342,491031552,634138750,810421248

%N Number of different fixed (possibly) disconnected pentominoes bounded tightly by an n X n square.

%H G. C. Greubel, <a href="/A163435/b163435.txt">Table of n, a(n) for n = 1..1000</a>

%H <a href="/index/Rec#order_07">Index entries for linear recurrences with constant coefficients</a>, signature (7,-21,35,-35,21,-7,1).

%F a(n) = 2/3*n^2*(n-2)^2*(5*n^2-10*n+2), n>1.

%F G.f.: 2*x^3*(51+539*x+574*x^2+30*x^3+7*x^4-x^5)/(1-x)^7. - _Colin Barker_, Apr 25 2012

%F E.g.f.: (2/3)*x*(5*x^5 + 45*x^4 + 87*x^3 + 24*x^2 + 3*x - 3)*exp(x) + 2*x. - _G. C. Greubel_, Dec 23 2016

%e a(3) = 102: there are 102 rotations of the 19 free (possibly) disconnected pentominoes bounded tightly by a 3 X 3 square; these include the F, T, V, W, X and Z (connected) pentominoes and 13 strictly disconnected free pentominoes.

%t Join[{0}, Table[(2/3)*n^2*(n - 2)^2*(5*n^2 - 10*n + 2), {n, 2, 50}]] (* or *) Join[{0}, LinearRecurrence[{7,-21,35,-35,21,-7,1}, {0, 102, 1792, 11550, 46848, 144550, 371712}, 50]] (* _G. C. Greubel_, Dec 23 2016 *)

%o (PARI) concat([0,0], Vec(2*x^3*(51+539*x+574*x^2+30*x^3+7*x^4-x^5)/ (1-x)^7 + O(x^50))) \\ _G. C. Greubel_, Dec 23 2016

%Y Cf. A162675, A163434, A163437.

%K nonn,easy

%O 1,3

%A _David Bevan_, Jul 28 2009

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Last modified July 5 21:00 EDT 2020. Contains 335473 sequences. (Running on oeis4.)