OFFSET
0,3
COMMENTS
If only symmetries of the rectangle are considered then a(4) would be 18.
LINKS
John Mason, Table of n, a(n) for n = 0..1000
John Mason, Counting free tilings of a rectangle
Index entries for linear recurrences with constant coefficients, signature (3,5,-13,-9,3,3,27,18,6,0,-12,-6,-12,-6,0,1,3,1,1).
FORMULA
For even n > 4:
2 * Sum_{k=0..(n - 2) / 2} (A054856(k))) / 4;
For odd n > 4:
Where compo(n) is the number of distinct compositions of n as a sum of 1, 2, (1+1) and 4.
G.f.: -5*x^4 + (1 - 2*x - 4*x^2 + 2*x^3 + 2*x^4 + 13*x^5 + 16*x^6 + x^7 - 5*x^8 - 13*x^9 - 7*x^10 - 2*x^11 + 3*x^13 + x^14 + x^15)/((1 + x)*(1 + x^2)*(1 - 3*x + x^4)*(1 - x - 2*x^2 - x^4)*(1 - 3*x^2 + x^8)). - Andrew Howroyd, Nov 16 2025
EXAMPLE
a(3) is 6 because of:
+-+-+-+ +-+-+-+ +-+-+-+ +-+-+-+ +-+-+-+ +-+-+-+
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+-+-+-+ + + + +-+ + +-+ + +-+ +-+-+-+
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+-+-+-+ + + +-+-+-+ +-+-+-+ +-+-+-+ + +-+
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+-+-+-+ +-+-+-+ + +-+ +-+ + +-+-+-+ +-+-+-+
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+-+-+-+ +-+-+-+ +-+-+-+ +-+-+-+ +-+-+-+ +-+-+-+
PROG
(PARI) Vec(-5*x^4+(1-2*x-4*x^2+2*x^3+2*x^4+13*x^5+16*x^6+x^7-5*x^8-13*x^9-7*x^10-2*x^11+3*x^13+x^14+x^15)/((1+x)*(1+x^2)*(1-3*x+x^4)*(1-x-2*x^2-x^4)*(1-3*x^2+x^8))+O(x^99)) \\ Charles R Greathouse IV, May 13 2026
CROSSREFS
Column k = 4 of A227690.
Sequences for fixed and free (inequivalent) tilings of m X n rectangles, for 2 <= m <= 10:
KEYWORD
nonn,easy
AUTHOR
John Mason, Dec 12 2022
STATUS
approved