%I #46 Sep 02 2024 10:41:47

%S -4,5,6,17,40,97,234,565,1364,3293,7950,19193,46336,111865,270066,

%T 651997,1574060,3800117,9174294,22148705,53471704,129092113,311655930,

%U 752403973,1816463876,4385331725,10587127326,25559586377,61706300080,148972186537,359650673154

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

%C From _Greg Dresden_, May 08 2023: (Start)

%C 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.

%C ._ _

%C |_|_|_ _ _ _ _

%C |_|_|_|_|_|_|_|

%C (End)

%H Reinhard Zumkeller, <a href="/A221174/b221174.txt">Table of n, a(n) for n = 0..1000</a>

%H José L. Ramírez, Gustavo N. Rubiano, and Rodrigo de Castro, <a href="http://arxiv.org/abs/1212.1368">A Generalization of the Fibonacci Word Fractal and the Fibonacci Snowflake</a>, arXiv preprint arXiv:1212.1368 [cs.DM], 2012-2014.

%H <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (2,1).

%F a(n) = 13*A000129(n) - 4*A000129(n+1). - _R. J. Mathar_, Jan 14 2013

%F G.f.: -(13*x-4) / (x^2+2*x-1). - _Colin Barker_, Jul 10 2015

%F 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

%t LinearRecurrence[{2, 1}, {-4, 5}, 50] (* _Paolo Xausa_, Sep 02 2024 *)

%o (Haskell)

%o a221174 n = a221174_list !! n

%o a221174_list = -4 : 5 : zipWith (+)

%o (map (* 2) $ tail a221174_list) a221174_list

%o -- _Reinhard Zumkeller_, Jan 04 2013

%o (PARI) Vec(-(13*x-4)/(x^2+2*x-1) + O(x^50)) \\ _Colin Barker_, Jul 10 2015

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

%K sign,easy

%O 0,1

%A _N. J. A. Sloane_, Jan 04 2013