A386489 Expansion of (1-x)/((1+x+2*x^2)*(1-4*x+x^2)).

1, 2, 7, 30, 109, 402, 1511, 5638, 21021, 78474, 292887, 1093006, 4079181, 15223810, 56815879, 212039702, 791343293, 2953333114, 11021988791, 41134623134, 153516503405, 572931388658, 2138209053735, 7979904827430, 29781410249821, 111145736175722

Offset: 0

Author

Greg Dresden and Madison Lingchen Zhou, Aug 20 2025

Keywords

• nonn
• easy
• new

Comments

a(n) is the number of ways to tile a 2 X n board with squares, dominoes, and L-shaped quadrominoes. Here is one of the a(4)=109 possible tilings of a 2 X 4 board:

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Compare to A030186 which counts the tilings with just squares and dominos.

Links

• Index entries for linear recurrences with constant coefficients, signature (3,1,7,-2).

Crossrefs

Cf. A030186, A033505.

Programs

Mathematica

LinearRecurrence[{3, 1, 7, -2}, {1, 2, 7, 30}, 30]

Formula

a(n) = 3*a(n-1) + a(n-2) + 7*a(n-3) - 2*a(n-4).

a(n) = A030186(n) + 2*sum_{i=0..n-2}(A033505(n-i-3)*a(i) + A030186(n-i-3)*(a(i)+2*sum_{j=0..i} a(j)).

a(n) ~ (2 + sqrt(3))^(n+2) / (18 + 4*sqrt(3)). - Vaclav Kotesovec, Aug 21 2025

23*a(n) = -4*A001353(n)+13*A001353(n+1) +10*A001607(n+1)+8*A001607(n) . - R. J. Mathar, Aug 26 2025