A163435 Number of different fixed (possibly) disconnected pentominoes bounded tightly by an n X n square.
0, 0, 102, 1792, 11550, 46848, 144550, 371712, 838782, 1715200, 3247398, 5779200, 9774622, 15843072, 24766950, 37531648, 55357950, 79736832, 112466662, 155692800, 211949598, 284204800, 375906342, 491031552, 634138750, 810421248
Offset: 1
Examples
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.
Links
- G. C. Greubel, Table of n, a(n) for n = 1..1000
- Index entries for linear recurrences with constant coefficients, signature (7,-21,35,-35,21,-7,1).
Programs
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Mathematica
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 *) -
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
Formula
a(n) = 2/3*n^2*(n-2)^2*(5*n^2-10*n+2), n>1.
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
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