A332647 a(n) = 2*a(n-1) + a(n-3) with a(0) = 3, a(1) = 2, a(2) = 4.

3, 2, 4, 11, 24, 52, 115, 254, 560, 1235, 2724, 6008, 13251, 29226, 64460, 142171, 313568, 691596, 1525363, 3364294, 7420184, 16365731, 36095756, 79611696, 175589123, 387274002, 854159700, 1883908523, 4155091048, 9164341796, 20212592115, 44580275278, 98324892352

Offset: 0

Author

Greg Dresden, Feb 18 2020

Keywords

• easy
• nonn

Comments

a(n) is the number of ways to tile a bracelet of length n with black trominos, and black or white squares.

Links

• Colin Barker, Table of n, a(n) for n = 0..1000
• Greg Dresden and Michael Tulskikh, Tilings of 2 X n boards with dominos and L-shaped trominos, Washington & Lee University (2021).
• Helmut Prodinger, On third-order Pell polynomials, arXiv:2011.04388 [math.NT], 2020.
• Index entries for linear recurrences with constant coefficients, signature (2,0,1).

Crossrefs

Cf. A008998, A052980. Equals one more than A080204.

Programs

Magma

a:=[3,2,4]; [n le 3 select a[n] else 2*Self(n-1)+Self(n-3):n in [1..33]]; // Marius A. Burtea, Feb 18 2020

Mathematica

LinearRecurrence[{2, 0, 1}, {3, 2, 4}, 50]

PARI

Vec((3 - 4*x) / (1 - 2*x - x^3) + O(x^30)) \\ Colin Barker, Feb 18 2020

PARI

polsym(x^3-2*x^2-1, 44) \\ Joerg Arndt, May 28 2020

Formula

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

a(n) = w1^n + w2^n + w3^n where w1,w2,w3 are the three roots of x^3-2x^2-1=0.

For n>2, a(n) = round(w1^n) for w1 the single real root of x^3-2x^2-1=0.

G.f.: (3 - 4*x) / (1 - 2*x - x^3). - Colin Barker, Feb 18 2020

a(n) = A008998(n) + 2*A008998(n-3) = 3*A008998(n) - 4*A008998(n-1).

a(n) = (5*b(n) - b(n-1) - b(n-2))/2 where b(n) = A052980(n). - Greg Dresden, Mar 10 2020

a(n) = A080204(n) + 1. - Greg Dresden, May 27 2020