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

Author

Alois P. Heinz, Apr 17 2013

Keywords

• nonn
• easy

Examples

			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)].
		

Links

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

Crossrefs

Column k=5 of A224697.

Programs

Magma

m:=50; R:=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

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);

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).