A097075 - OEIS

%I #33 Apr 01 2024 04:14:07

%S 1,0,2,3,9,20,50,119,289,696,1682,4059,9801,23660,57122,137903,332929,

%T 803760,1940450,4684659,11309769,27304196,65918162,159140519,

%U 384199201,927538920,2239277042,5406093003,13051463049,31509019100,76069501250

%N Expansion of g.f. (1-x-x^2)/(1-x-3*x^2-x^3).

%C Counts closed walks of length n at a vertex of a triangle, to which a loop has been added at one of the other vertices.

%C a(n) is the top left entry of the n-th power of the 3 X 3 matrix [0, 1, 1; 1, 1, 1; 1, 1, 0] or of the 3 X 3 matrix [0, 1, 1; 1, 0, 1; 1, 1, 1].

%H G. C. Greubel, <a href="/A097075/b097075.txt">Table of n, a(n) for n = 0..1000</a>

%H J. Bodeen, S. Butler, T. Kim, X. Sun, and S. Wang, <a href="https://doi.org/10.37236/3478">Tiling a strip with triangles</a>, El. J. Combinat. 21 (1) (2014) P1.7.

%H <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (1,3,1).

%F a(n) = ((1+sqrt(2))^n + (1-sqrt(2))^n + 2*(-1)^n)/4.

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

%F a(n) = (1/2)*((-1)^n + Sum_{k=0..floor(n/2)} binomial(n, 2*k)*2^k).

%F a(n) = ((-1)^n + A001333(n))/2.

%F E.g.f.: (cosh(x) + exp(x)*cosh(sqrt(2)*x) - sinh(x))/2. - _Stefano Spezia_, Mar 31 2024

%t LinearRecurrence[{1,3,1}, {1,0,2}, 41] (* or *) Table[(LucasL[n,2] +2*(-1)^n)/4, {n,0,40}] (* _G. C. Greubel_, Aug 18 2022 *)

%o (PARI) Vec((1-x-x^2)/(1-x-3*x^2-x^3) + O(x^50)) \\ _Michel Marcus_, Mar 25 2014

%o (Magma) [(Evaluate(DicksonFirst(n,-1), 2) +2*(-1)^n)/4: n in [0..40]]; // _G. C. Greubel_, Aug 18 2022

%o (SageMath) [(lucas_number2(n,2,-1) +2*(-1)^n)/4 for n in (0..40)] # _G. C. Greubel_, Aug 18 2022

%Y Cf. A000129, A001333, A051927, A097076.

%K easy,nonn

%O 0,3

%A _Paul Barry_, Jul 22 2004