cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

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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].

Original entry on oeis.org

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

Views

Author

Gary W. Adamson, May 02 2004

Keywords

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

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

A094432 a(n) = rightmost term in M^n * [1 0 0]. M = the 3 X 3 stiffness matrix [1 -1 0 / -1 4 -3 / 0 -3 3].

Original entry on oeis.org

0, 3, 24, 165, 1104, 7347, 48840, 324597, 2157216, 14336355, 95275896, 633179973, 4207956720, 27965034003, 185848661544, 1235103986325, 8208193936704, 54549615616707, 362523179503320, 2409238895476197, 16011202548279696
Offset: 1

Views

Author

Gary W. Adamson, May 02 2004

Keywords

Comments

A094431(n) = left term in M^n * [1 0 0]. A stiffness matrix in Hooke's Law governs the force on nodes of stretched or compressed springs (refer to A094431). a(n)/a(n-1) tends to 4 + sqrt(7) = 6.6457513...; a(n)/A094431(n) tends to 2 + sqrt(7). A stiffness matrix is symmetric.

Examples

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

References

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

Links

Crossrefs

Cf. A094431, A154245.

Programs

  • Mathematica
    Table[(MatrixPower[{{1, -1, 0}, {-1, 4, -3}, {0, -3, 3}}, n].{1, 0, 0})[[3]], {n, 21}] (* Robert G. Wilson v, May 08 2004 *)

Formula

a(n) = (3/(2*sqrt(7)))*((4+sqrt(7))^(n-1)-(4-sqrt(7))^(n-1)). For n>1, a(n) = 3*A154245(n-1). - Francesco Daddi, Aug 02 2011
G.f.: 3*x^2/(1-8*x+9*x^2). - Bruno Berselli, Aug 03 2011

Extensions

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

A094434 a(n) = rightmost term of M^n * [1 0 0], with M = the 3 X 3 matrix [1 -1 0 / -1 3 -2 / 0 -2 2].

Original entry on oeis.org

0, 2, 12, 60, 288, 1368, 6480, 30672, 145152, 686880, 3250368, 15380928, 72783360, 344414592, 1629787392, 7712236800, 36494696448, 172694757888, 817200368640, 3867033664512, 18298999775232, 86591796664320, 409756781334528
Offset: 1

Views

Author

Gary W. Adamson, May 02 2004

Keywords

Comments

Left term in M^n * [1 0 0] = A094433(n). a(n)/ a(n-1) tends to 3 + sqrt(3) = 4.732050807...; e.g. a(9)/a(8) = 145152/30672 = 4.732394... 3. a(n)/ A094433(n) tends to 1 + sqrt(3); e.g. a(9)/A094433(9) = 145152/53136 = 2.731707... 4. M = a "stiffness matrix" with k1 = 1, k2 = 2, relating to Hooke's law governing the force on the nodes of compressed or stretched springs with stiffness constants k1, k2. (see A094433, A094431).

Examples

			a(4) = 60 since M^4 * [1 0 0] = [24 -84 60].

References

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

Links

Crossrefs

Cf. A094431, A094431, A094433.

Programs

  • Mathematica
    Table[(MatrixPower[{{1, -1, 0}, {-1, 3, -2}, {0, -2, 2}}, n].{1, 0, 0})[[3]], {n, 24}] (* Robert G. Wilson v *)
LinearRecurrence[{6,-6},{0,2},30] (* Harvey P. Dale, May 01 2017 *)

Formula

a(n) = 6*a(n-1)-6*a(n-2). G.f.: 2*x^2/(1-6*x+6*x^2). [Colin Barker, Sep 05 2012]

Extensions

More terms from Robert G. Wilson v, May 08 2004
Showing 1-3 of 3 results.

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