A166482 a(n) = Sum_{k=0..n} binomial(n+k,2k)*Fibonacci(2k+1).

1, 3, 12, 51, 221, 965, 4227, 18540, 81363, 357145, 1567849, 6883059, 30218028, 132664227, 582428789, 2557009709, 11225925267, 49284687948, 216372426339, 949930508209, 4170438905425, 18309298027683, 80382521554380

Offset: 0

Author

Paul Barry, Oct 14 2009

Keywords

• easy
• nonn

Comments

Conjecture: a(n) is the number of tilings of a 4 X 4n rectangle into L tetrominoes (no reflections, only rotations). - Nicolas Bělohoubek, Feb 12 2022

The conjecture above was confirmed by Nicolas Bělohoubek and Antonín Slavík. (See links.) - Nicolas Bělohoubek, Jan 21 2025

Links

• G. C. Greubel, Table of n, a(n) for n = 0..500
• Nicolas Bělohoubek and Antonín Slavík, L-Tetromino Tilings and Two-Color Integer Compositions, Univ. Karlova (Czechia, 2025). See p. 5.
• Index entries for linear recurrences with constant coefficients, signature (7,-13,7,-1).

Crossrefs

Cf. A131322.

Programs

Mathematica

CoefficientList[Series[(1-4x+4x^2-x^3)/(1-7x+13x^2-7x^3+x^4), {x,0,30}],x]  (* Harvey P. Dale, Mar 23 2011 *)
LinearRecurrence[{7, -13, 7, -1}, {1, 3, 12, 51}, 50] (* G. C. Greubel, May 15 2016 *)

Formula

G.f.: (1 - 4x + 4x^2 - x^3)/(1 - 7x + 13x^2 - 7x^3 + x^4).

a(n) = Sum_{k=0..n} binomial(n+k,2k) * Sum_{j=0..k} binomial(k+j,2j).

a(n) ~ (1 + 1/sqrt(5) + 2*sqrt(31/290 + 13/(58*sqrt(5)))) * ((7 + sqrt(5) + sqrt(38 + 14*sqrt(5)))^n / 2^(2*n+2)). - Vaclav Kotesovec, Feb 22 2022

a(n) = A131322(2*n). - Nicolas Bělohoubek, Jan 21 2025