A131322 Row sums of triangle A131321.

1, 1, 3, 5, 12, 23, 51, 103, 221, 456, 965, 2009, 4227, 8833, 18540, 38803, 81363, 170399, 357145, 748176, 1567849, 3284833, 6883059, 14421533, 30218028, 63314735, 132664227, 277968871, 582428789, 1220356440, 2557009709

Offset: 0

Author

Gary W. Adamson, Jun 28 2007

Keywords

• nonn

Comments

Equals INVERT transform of (1, 2, 0, 1, 0, 1, 0, 1, ...). - Gary W. Adamson, Apr 28 2009

The sequence is also the INVERT transform of the aerated odd-indexed Fibonacci numbers (i.e., of (1, 0, 2, 0, 5, 0, ...)). Sequence A124400 is the INVERT transform of the aerated even-indexed Fibonacci numbers. - Gary W. Adamson, Feb 07 2014

a(n) is the number of tilings of a 4 X 2n rectangle into L tetrominoes (no reflections, only rotations). - Nicolas Bělohoubek, Jan 21 2025

Examples

			a(4) = 12 = 5 + 0 + 6 + 0 + 1.
		

Links

• Paolo Xausa, Table of n, a(n) for n = 0..1000
• Nicolas Bělohoubek and Antonín Slavík, L-Tetromino Tilings and Two-Color Integer Compositions, Univ. Karlova (Czechia, 2025). See p. 3.
• Index entries for linear recurrences with constant coefficients, signature (1,3,-1,-1).

Crossrefs

Cf. A166482, A131321, A000045.

Cf. A124400.

Programs

Mathematica

LinearRecurrence[{1, 3, -1, -1}, {1, 1, 3, 5}, 50] (* Paolo Xausa, Jan 28 2025 *)

Formula

G.f.: (1-x^2)/(1 - x - 3x^2 + x^3 + x^4). - Philippe Deléham, Jan 21 2012

a(n) = a(n-1) + 3*a(n-2) - a(n-3) - a(n-4), a(0)=1, a(1)=1, a(2)=3, a(3)=5. - Philippe Deléham, Jan 21 2012

a(n) = Sum_{m=0..ceiling(n/2)} binomial(n-m,n-2*m)*Fibonacci(n-2*m+1). - Vladimir Kruchinin, Jan 26 2013

From Nicolas Bělohoubek, Jan 21 2025: (Start)

a(n) = Sum_{m=1..4} (alpha_m * x_m^n). For x_m and alpha_m values see "L-tetromino tilings" article in links.

a(2*n) = A166482(n). (End)

Extensions

a(10)-a(30) from Philippe Deléham, Jan 21 2012