A094432 -id:A094432 - cp's OEIS frontend

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

Gary W. Adamson, May 02 2004

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

			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 + ...

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

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).

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

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

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 *)

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

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

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

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

A094431 a(n) = left term in M^n * [1 0 0], where M = the 3 X 3 matrix [1 -1 0 / -1 4 -3 / 0 -3 3].

Original entry on oeis.org

1, 1, 2, 7, 38, 241, 1586, 10519, 69878, 464353, 3085922, 20508199, 136292294, 905764561, 6019485842, 40004005687, 265856672918, 1766817332161, 11741828601026, 78033272818759, 518589725140838, 3446418345757873, 22904039239795442, 152214548806542679, 1011580037294182454, 6722709359094575521

Offset: 0

Gary W. Adamson, May 02 2004

a(n)/a(n-1) tends to 4 + sqrt(7) = 6.6457513... A094432(n)/a(n) tends to 2 + sqrt(7) = 4.645638... 3. M is a "stiffness matrix" K = [k1 -k1 0 / -k1 (k1 + k2) -k2 / 0 -k2 k2] with k1 = 1, k2 = 3. K governs the force exerted on a spring with nodes, in comparison with the spring in a "no tension" position (Fig 3.2.1, p. 86, Meyer). "Stretching or compressing the springs creates a force on each node according to Hooke's law that says that the force exerted by a spring is F = kx where x is the distance the spring is stretched or compressed and where k is the stiffness constant inherent to the spring".

			a(4) = 38 since M^4 * [1 0 0] =[38 -203 165].

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

Andrew Howroyd, Table of n, a(n) for n = 0..500
Index entries for linear recurrences with constant coefficients, signature (8,-9).

Cf. A094432.

Table[(MatrixPower[{{1, -1, 0}, {-1, 4, -3}, {0, -3, 3}}, n].{1, 0, 0})[[1]], {n, 21}] (* Robert G. Wilson v *)

From Colin Barker, Apr 02 2012: (Start)
a(n) = 8*a(n-1) - 9*a(n-2).
G.f.: (1 - 7*x + 3*x^2)/(1 - 8*x + 9*x^2). (End)

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

a(0)=1 prepended by Andrew Howroyd, Dec 27 2024

A154245 a(n) = ( (4 + sqrt(7))^n - (4 - sqrt(7))^n )/(2*sqrt(7)).

Original entry on oeis.org

1, 8, 55, 368, 2449, 16280, 108199, 719072, 4778785, 31758632, 211059991, 1402652240, 9321678001, 61949553848, 411701328775, 2736064645568, 18183205205569, 120841059834440, 803079631825399, 5337067516093232

Offset: 1

Al Hakanson (hawkuu(AT)gmail.com), Jan 05 2009

Second binomial transform of A109115.

Lim_{n -> infinity} a(n)/a(n-1) = 4 + sqrt(7) = 6.6457513110....

Vincenzo Librandi, Table of n, a(n) for n = 1..100
Index entries for linear recurrences with constant coefficients, signature (8,-9).

Equals (A094432 without initial term 0)/3.

Cf. A010465 (decimal expansion of square root of 7), A109115.

a:=[1,8];; for n in [3..30] do a[n]:=8*a[n-1]-9*a[n-2]; od; a; # G. C. Greubel, May 21 2019

Z:=PolynomialRing(Integers()); N:=NumberField(x^2-7); S:=[ ((4+r)^n-(4-r)^n)/(2*r): n in [1..20] ]; [ Integers()!S[j]: j in [1..#S] ]; // Klaus Brockhaus, Jan 07 2009

I:=[1,8]; [n le 2 select I[n] else 8*Self(n-1)-9*Self(n-2): n in [1..30]]; // Vincenzo Librandi, Sep 08 2016

Table[Simplify[((4+Sqrt[7])^n -(4-Sqrt[7])^n)/(2*Sqrt[7])], {n, 30}] (* or *) LinearRecurrence[{8, -9},{1, 8}, 30] (* G. C. Greubel, Sep 07 2016 *)
Rest@ CoefficientList[Series[x/(1 -8x +9x^2), {x, 0, 30}], x] (* Michael De Vlieger, Sep 08 2016 *)

my(x='x+O('x^30)); Vec( x/(1-8*x+9*x^2) ) \\ G. C. Greubel, May 21 2019

[lucas_number1(n,8,9) for n in range(1, 21)] # Zerinvary Lajos, Apr 23 2009

From Philippe Deléham, Jan 06 2009: (Start)
a(n) = 8*a(n-1) - 9*a(n-2) for n > 1, with a(0)=0, a(1)=1.
G.f.: x/(1 - 8*x + 9*x^2). (End)
a(n) = b such that (3^(n-1)/2)*Integral_{x=0..Pi/2} (sin(n*x))/(4/3-cos(x)) dx = c + b*log(2). - Francesco Daddi, Aug 02 2011
E.g.f.: (1/sqrt(7))*exp(4*x)*sinh(sqrt(7)*x). - G. C. Greubel, Sep 07 2016

Extended beyond a(7) by Klaus Brockhaus, Jan 07 2009

Edited by Klaus Brockhaus, Oct 06 2009

cp's OEIS Frontend

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].

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A094431 a(n) = left term in M^n * [1 0 0], where M = the 3 X 3 matrix [1 -1 0 / -1 4 -3 / 0 -3 3].

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A154245 a(n) = ( (4 + sqrt(7))^n - (4 - sqrt(7))^n )/(2*sqrt(7)).

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