A187753 Number of different ways to divide an n X 5 rectangle into subsquares, considering only the list of parts.

1, 1, 3, 5, 9, 11, 20, 26, 36, 48, 64, 80, 106, 128, 160, 195, 238, 281, 340, 397, 467, 544, 633, 724, 838, 950, 1083, 1226, 1385, 1550, 1745, 1942, 2165, 2402, 2663, 2933, 3242, 3555, 3902, 4270, 4667, 5079, 5539, 6007, 6518, 7055, 7631, 8227, 8880, 9547

Offset

0,3

Links

Alois P. Heinz, Table of n, a(n) for n = 0..1000

Formula

G.f.: (x^9 - x^8 + x^7 - 4*x^6 + x^5 - 3*x^4 - x^3 - 2*x^2 - 1) / (x^19 - x^18 - x^16 + 2*x^12 + x^10 - x^9 - 2*x^7 + x^3 + x - 1).

Example

a(4) = 9 because there are 9 ways to divide a 4 X 5 rectangle into subsquares, considering only the list of parts: [20(1 X 1)], [16(1 X 1), 1(2 X 2)], [12(1 X 1), 2(2 X 2)], [11(1 X 1), 1(3 X 3)], [8(1 X 1), 3(2 X 2)], [7(1 X 1), 1(2 X 2), 1(3 X 3)], [4(1 X 1), 4(2 X 2)], [4(1 X 1), 1(4 X 4)], [3(1 X 1), 2(2 X 2), 1(3 X 3)].  There is no way to divide this rectangle into [2(1 X 1), 2(3 X 3)].

Maple

gf:= (x^9-x^8+x^7-4*x^6+x^5-3*x^4-x^3-2*x^2-1)/
     (x^19-x^18-x^16+2*x^12+x^10-x^9-2*x^7+x^3+x-1):
a:= n-> coeff(series(gf, x, n+1), x, n):
seq(a(n), n=0..60);

Prog

(Magma) m:=50; R<x>:=PowerSeriesRing(Integers(), m); Coefficients(R!((1+2*x^2+x^3+3*x^4-x^5+4*x^6-x^7+x^8-x^9)/((1-x)^5*(1+x)^2*(1+x^2)*(1-x +x^2)*(1+x+x^2)^2*(1+x+x^2+x^3+x^4)))); // Bruno Berselli, Apr 17 2013

Crossrefs

Column k=5 of A224697.

Sequence in context: A070639 A078392 A364808 * A339638 A231716 A113488

Adjacent sequences: A187750 A187751 A187752 * A187754 A187755 A187756

Keyword

nonn,easy

Author

Alois P. Heinz, Apr 17 2013

Status

approved