A187753 - OEIS

%I #33 Sep 08 2022 08:45:56

%S 1,1,3,5,9,11,20,26,36,48,64,80,106,128,160,195,238,281,340,397,467,

%T 544,633,724,838,950,1083,1226,1385,1550,1745,1942,2165,2402,2663,

%U 2933,3242,3555,3902,4270,4667,5079,5539,6007,6518,7055,7631,8227,8880,9547

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

%H Alois P. Heinz, <a href="/A187753/b187753.txt">Table of n, a(n) for n = 0..1000</a>

%F 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).

%e 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)].

%p gf:= (x^9-x^8+x^7-4*x^6+x^5-3*x^4-x^3-2*x^2-1)/

%p      (x^19-x^18-x^16+2*x^12+x^10-x^9-2*x^7+x^3+x-1):

%p a:= n-> coeff(series(gf, x, n+1), x, n):

%p seq(a(n), n=0..60);

%o (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

%Y Column k=5 of A224697.

%K nonn,easy

%O 0,3

%A _Alois P. Heinz_, Apr 17 2013