%I #49 Apr 20 2022 03:25:04
%S 1,1,2,6,24,108,504,2376,11232,53136,251424,1189728,5629824,26640576,
%T 126064512,596543616,2822874624,13357986048,63210668544,299116094976,
%U 1415432558592,6697898781696,31694797338624,149981391341568,709719564017664,3358429036056576
%N a(n) is the left term in M^n * [1 0 0], M = the 3 X 3 matrix [1 -1 0 / -1 3 -2 / 0 -2 2].
%C Right term of M^n * [1 0 0] = A094434(n).
%C a(n)/a(n-1) tends to 3 + sqrt(3) = 4.732050807... (A165663).
%C A094434(n)/a(n) tends to 1 + sqrt(3) = 2.732050807... (A090388).
%C M is a "stiffness matrix" with k1 = 1, k2 = 2; in K = [k1 -k1 0 / -k1 (k1 + k2) -k2 / 0 -k2 k2], where K relates to Hooke's Law governing the force on nodes of springs resulting from stretching or compressing the springs (see A094431).
%C The eigenvalues of M are 3+sqrt(3), 3-sqrt(3) and 0. - Tamas Kalmar-Nagy (integers(AT)kalmarnagy.com), Mar 23 2008
%C a(n) is the number of permutations of length n+1 avoiding the partially ordered pattern (POP) {1>2, 1>3, 1>4, 5>2, 5>3, 5>4} of length 5. That is, the number of length n+1 permutations having no subsequences of length 5 in which the elements in positions 1 and 5 are larger than the elements in positions 2, 3 and 4. - _Sergey Kitaev_, Dec 11 2020
%D Carl D. Meyer, "Matrix Analysis and Applied Linear Algebra", SIAM, 2000, p. 86-87.
%H Michael De Vlieger, <a href="/A094433/b094433.txt">Table of n, a(n) for n = 0..1483</a>
%H Alice L. L. Gao and Sergey Kitaev, <a href="https://arxiv.org/abs/1903.08946">On partially ordered patterns of length 4 and 5 in permutations</a>, arXiv:1903.08946 [math.CO], 2019.
%H Alice L. L. Gao and Sergey Kitaev, <a href="https://doi.org/10.37236/8605">On partially ordered patterns of length 4 and 5 in permutations</a>, The Electronic Journal of Combinatorics 26(3) (2019), P3.26.
%H Sergey Kitaev and Artem Pyatkin, <a href="https://arxiv.org/abs/2204.08936">On permutations avoiding partially ordered patterns defined by bipartite graphs</a>, arXiv:2204.08936 [math.CO], 2022.
%H Takao Komatsu, <a href="https://arxiv.org/abs/2105.08277">Asymmetric Circular Graph with Hosoya Index and Negative Continued Fractions</a>, arXiv:2105.08277 [math.CO], 2021.
%H <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (6,-6).
%F a(n) = (3+sqrt(3))^(n-2) + (3-sqrt(3))^(n-2). - Tamas Kalmar-Nagy (integers(AT)kalmarnagy.com), Mar 23 2008 [Corrected by _R. J. Mathar_, Mar 28 2010, Jun 02 2010]
%F G.f.: 1 + x*(1-4*x)/(1-6*x+6*x^2). - _R. J. Mathar_, Mar 28 2010
%e a(4) = 24 since M^4 * [1 0 0] = [24 -84 60].
%e G.f. = 1 + x + 2*x^2 + 6*x^3 + 24*x^4 + 108*x^5 + 504*x^6 + 2376*x^7 + ...
%p a:= n-> (<<1|-1|0>, <-1|3|-2>, <0|-2|2>>^n)[1$2]:
%p seq(a(n), n=0..28); # _Alois P. Heinz_, Dec 11 2020
%t Table[(MatrixPower[{{1, -1, 0}, {-1, 3, -2}, {0, -2, 2}}, n].{1, 0, 0})[[1]], {n, 24}] (* _Robert G. Wilson v_ *)
%t Table[(3 + Sqrt[3])^n + (3 - Sqrt[3])^n, {n, 0, 20}] // Simplify (* Tamas Kalmar-Nagy (integers(AT)kalmarnagy.com), Mar 23 2008 *)
%t Rest@ CoefficientList[Series[x (1 - 4 x)/(1 - 6 x + 6 x^2), {x, 0, 23}], x] (* _Michael De Vlieger_, May 01 2019 *)
%o (Sage) [lucas_number2(n,6,6)for n in range(-1,23)] # _Zerinvary Lajos_, Jul 08 2008
%Y Cf. A094431, A094432, A094434, A165663, A090388.
%K nonn,easy
%O 0,3
%A _Gary W. Adamson_, May 02 2004
%E More terms from _Robert G. Wilson v_, May 08 2004
%E a(0)=1 prepended by _Alois P. Heinz_, Dec 11 2020