A221174 a(0)=-4, a(1)=5; thereafter a(n) = 2*a(n-1) + a(n-2).

-4, 5, 6, 17, 40, 97, 234, 565, 1364, 3293, 7950, 19193, 46336, 111865, 270066, 651997, 1574060, 3800117, 9174294, 22148705, 53471704, 129092113, 311655930, 752403973, 1816463876, 4385331725, 10587127326, 25559586377, 61706300080, 148972186537, 359650673154

Offset: 0

Author

N. J. A. Sloane, Jan 04 2013

Keywords

• sign
• easy

Comments

From Greg Dresden, May 08 2023: (Start)

For n >= 3, 2*a(n) is the number of ways to tile this figure of length n-1 with two colors of squares and one color of domino. For n=8, we have here the figure of length n-1=7, and it has 2*a(8) = 2728 different tilings.

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(End)

Links

• Reinhard Zumkeller, Table of n, a(n) for n = 0..1000
• José L. Ramírez, Gustavo N. Rubiano, and Rodrigo de Castro, A Generalization of the Fibonacci Word Fractal and the Fibonacci Snowflake, arXiv preprint arXiv:1212.1368 [cs.DM], 2012-2014.
• Index entries for linear recurrences with constant coefficients, signature (2,1).

Crossrefs

Cf. A000129, A078343, A221172, A221173, A221175.

Programs

Haskell

a221174 n = a221174_list !! n
a221174_list = -4 : 5 : zipWith (+)
                        (map (* 2) $ tail a221174_list) a221174_list
-- Reinhard Zumkeller, Jan 04 2013

Mathematica

LinearRecurrence[{2, 1}, {-4, 5}, 50] (* Paolo Xausa, Sep 02 2024 *)

PARI

Vec(-(13*x-4)/(x^2+2*x-1) + O(x^50)) \\ Colin Barker, Jul 10 2015

Formula

a(n) = 13*A000129(n) - 4*A000129(n+1). - R. J. Mathar, Jan 14 2013

G.f.: -(13*x-4) / (x^2+2*x-1). - Colin Barker, Jul 10 2015

a(n) is the numerator of the continued fraction [4, 2, ..., 2, 4] with n-3 2's in the middle. For denominators, see A048654. - Greg Dresden and Tongjia Rao, Sep 02 2021