A094433 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].

1, 1, 2, 6, 24, 108, 504, 2376, 11232, 53136, 251424, 1189728, 5629824, 26640576, 126064512, 596543616, 2822874624, 13357986048, 63210668544, 299116094976, 1415432558592, 6697898781696, 31694797338624, 149981391341568, 709719564017664, 3358429036056576

Offset: 0

Author

Gary W. Adamson, May 02 2004

Keywords

• nonn
• easy

Comments

Right term of M^n * [1 0 0] = A094434(n).

a(n)/a(n-1) tends to 3 + sqrt(3) = 4.732050807... (A165663).

A094434(n)/a(n) tends to 1 + sqrt(3) = 2.732050807... (A090388).

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).

The eigenvalues of M are 3+sqrt(3), 3-sqrt(3) and 0. - Tamas Kalmar-Nagy (integers(AT)kalmarnagy.com), Mar 23 2008

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

Examples

			a(4) = 24 since M^4 * [1 0 0] = [24 -84 60].
G.f. = 1 + x + 2*x^2 + 6*x^3 + 24*x^4 + 108*x^5 + 504*x^6 + 2376*x^7 + ...
		

References

• Carl D. Meyer, "Matrix Analysis and Applied Linear Algebra", SIAM, 2000, p. 86-87.

Links

• Michael De Vlieger, Table of n, a(n) for n = 0..1483
• Alice L. L. Gao and Sergey Kitaev, On partially ordered patterns of length 4 and 5 in permutations, arXiv:1903.08946 [math.CO], 2019.
• Alice L. L. Gao and Sergey Kitaev, On partially ordered patterns of length 4 and 5 in permutations, The Electronic Journal of Combinatorics 26(3) (2019), P3.26.
• Sergey Kitaev and Artem Pyatkin, On permutations avoiding partially ordered patterns defined by bipartite graphs, arXiv:2204.08936 [math.CO], 2022.
• Takao Komatsu, Asymmetric Circular Graph with Hosoya Index and Negative Continued Fractions, arXiv:2105.08277 [math.CO], 2021.
• Index entries for linear recurrences with constant coefficients, signature (6,-6).

Crossrefs

Cf. A094431, A094432, A094434, A165663, A090388.

Programs

Maple

a:= n-> (<<1|-1|0>, <-1|3|-2>, <0|-2|2>>^n)[1$2]:
seq(a(n), n=0..28);  # Alois P. Heinz, Dec 11 2020

Mathematica

Table[(MatrixPower[{{1, -1, 0}, {-1, 3, -2}, {0, -2, 2}}, n].{1, 0, 0})[[1]], {n, 24}] (* Robert G. Wilson v *)
Table[(3 + Sqrt[3])^n + (3 - Sqrt[3])^n, {n, 0, 20}] // Simplify (* Tamas Kalmar-Nagy (integers(AT)kalmarnagy.com), Mar 23 2008 *)
Rest@ CoefficientList[Series[x (1 - 4 x)/(1 - 6 x + 6 x^2), {x, 0, 23}], x] (* Michael De Vlieger, May 01 2019 *)

Sage

[lucas_number2(n,6,6)for n in range(-1,23)] # Zerinvary Lajos, Jul 08 2008

Formula

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]

G.f.: 1 + x*(1-4*x)/(1-6*x+6*x^2). - R. J. Mathar, Mar 28 2010

Extensions

More terms from Robert G. Wilson v, May 08 2004

a(0)=1 prepended by Alois P. Heinz, Dec 11 2020