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%I #31 Apr 10 2024 10:44:51
%S 1,4,34,322,3164,31484,314662,3149674,31544384,315981452,3165414034,
%T 31710994234,317682195692,3182564368244,31883205466534,
%U 319408833724882,3199866987994304,32056562443839284,321145602837871522,3217266324544621714,32230871396722195484
%N Number of ways of dividing a 3 X n rectangle into rectangles of integer side lengths.
%H David A. Klarner and Spyros S. Magliveras, <a href="https://doi.org/10.1016/S0195-6698(88)80062-3">The number of tilings of a block with blocks</a>, European Journal of Combinatorics 9 (1988), 317-330.
%H J. Smith and H. Verrill, <a href="/A116694/a116694.pdf">On dividing rectangles into rectangles</a>.
%H <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (15,-55,51).
%F a(n) = 18*a(n-1) -100*a(n-2) +216*a(n-3) -153*a(n-4) with n>4 (see paper in Link lines, p. 1).
%F G.f.: 1+2*x*(2-13*x+16*x^2) / (1-15*x+55*x^2-51*x^3) = 1+2*x*(2-19*x+55*x^2-48*x^3) / (1-18*x+100*x^2-216*x^3+153*x^4). [_Bruno Berselli_, Apr 24 2012]
%F a(n) = 15*a(n-1) -55*a(n-2) +51*a(n-3) with n>3. [_Bruno Berselli_, Apr 24 2012]
%e For n=1 the a(1) = 4 ways to divide are:
%e ._ _ _ _
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%t Join[{1}, LinearRecurrence[{15, -55, 51}, {4, 34, 322}, 20]] (* _Bruno Berselli_, Apr 24 2012 *)
%Y Cf. A034999, A116694, A182275.
%K nonn,easy
%O 0,2
%A _Matthew C. Russell_, Apr 23 2012
%E More terms from _Bruno Berselli_, Apr 24 2012
%E a(0) added by _Alois P. Heinz_, Dec 10 2012
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