A208215 Number of ways of dividing a 3 X n rectangle into rectangles of integer side lengths.

1, 4, 34, 322, 3164, 31484, 314662, 3149674, 31544384, 315981452, 3165414034, 31710994234, 317682195692, 3182564368244, 31883205466534, 319408833724882, 3199866987994304, 32056562443839284, 321145602837871522, 3217266324544621714, 32230871396722195484

Offset

0,2

Links

Table of n, a(n) for n=0..20.

Pablo Blanco, Robert Dougherty-Bliss, Natalya Ter-Saakov, and Doron Zeilberger, In How Many Ways can a Rectangle be Rectangled?, arXiv:2606.05439 [math.CO], 2026. See p. 3.

David A. Klarner and Spyros S. Magliveras, The number of tilings of a block with blocks, Europ. J. Comb. 9 (1988), 317-330.

Joshua Smith and Helena Verrill, On dividing rectangles into rectangles, Louisiana State Univ. (2006).

Index entries for linear recurrences with constant coefficients, signature (15,-55,51).

Formula

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).

From Bruno Berselli, Apr 24 2012: (Start)

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).

a(n) = 15*a(n-1) -55*a(n-2) +51*a(n-3) with n>3. (End)

Limit_{n->oo} a(n+1)/a(n) = 10. - Stefano Spezia, Jun 10 2026

Example

For n=1 the a(1) = 4 ways to divide are:
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Mathematica

Join[{1}, LinearRecurrence[{15, -55, 51}, {4, 34, 322}, 20]] (* Bruno Berselli, Apr 24 2012 *)

Crossrefs

Cf. A034999, A116694, A182275.

Sequence in context: A005569 A370171 A232910 * A337390 A025572 A093137

Adjacent sequences: A208212 A208213 A208214 * A208216 A208217 A208218

Keyword

nonn,easy,changed

Author

Matthew C. Russell, Apr 23 2012

Extensions

More terms from Bruno Berselli, Apr 24 2012

a(0) added by Alois P. Heinz, Dec 10 2012

Status

approved