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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A141384 Trace of the n-th power of a certain 8X8 adjacency matrix.

Original entry on oeis.org

8, 8, 32, 158, 828, 4408, 23564, 126106, 675076, 3614144, 19349432, 103593806, 554625900, 2969386480, 15897666068, 85113810058, 455687062276, 2439682811480, 13061709929936, 69930511268510, 374397872321628
Offset: 0

Views

Author

Gerard P. Michon, Jun 29 2008

Keywords

Comments

a(n) is the trace of the n-th power of the adjacency matrix of order 8 whose rows are (up to simultaneous permutations of the rows and columns): 10111010 01111001 01111001 10111010 00111000 11111111 11111111 11111111.
For n>2, this is also the number of ways to mark one edge at every vertex of a regular n-gonal prism so that no edge is marked at both extremities.
Remarkably, for n>1, a(n)=A141221(n)+2.
The fourth-order linear recurrence established by Max Alekseyev for A141221, based on the minimal polynomial of the above (singular) matrix, namely x(x-1)(x^3-7x^2+9x-1) = x^5-8*x^4+16*x^3-10*x^2+x. Since its degree is 5, the corresponding recurrence holds for corresponding elements of the successive powers (or sums thereof, including matrix traces) only for n>=5. The recurrence would be valid down to n=4 if we had a(0)=4, which is not the case.

Examples

			a(0) = 8 because the trace of the order-8 identity matrix is 8.
a(1) = 8 because all diagonal elements of the adjacency matrix are 1 (there's a loop at each vertex).

Links

Crossrefs

Cf. A141221.

Formula

For n>=5, a(n) = 8*a(n-1)-16*a(n-2)+10*a(n-3)-a(n-4).
For positive values of n: a(n) = (5.3538557854308)^n + (1.5235479602692)^n + 1 + (0.1225962542999)^n. The dominant term in the above is the n-th power of (7+2*sqrt(22)*cos(atan(sqrt(5319)/73)/3))/3.
G.f.: 2*(4-28*x+48*x^2-25*x^3+2*x^4)/((1-x)*(1-7*x+9*x^2-x^3)). [Colin Barker, Jan 20 2012]

Extensions

Edited by Max Alekseyev, Aug 03 2015

A141385 a(n) = 7*a(n-1) - 9*a(n-2) + a(n-3) with a(0)=3, a(1)=7, a(2)=31.

Original entry on oeis.org

3, 7, 31, 157, 827, 4407, 23563, 126105, 675075, 3614143, 19349431, 103593805, 554625899, 2969386479, 15897666067, 85113810057, 455687062275, 2439682811479, 13061709929935, 69930511268509, 374397872321627
Offset: 0

Views

Author

Gerard P. Michon, Jul 02 2008, Jul 23 2008

Keywords

Comments

The old definition given for this sequence was "A sequence obeying a third-order linear recurrence".
Ruling out finitely many exceptional terms, this sequence differs by a constant from several related enumerations with a slightly more complicated structure (fourth-order linear recurrence):
For n>0, A141221(n) = a(n) - 1. For n>2, A141384(n) = a(n) + 1.

Examples

			a(0) = 3 = A^0 + B^0 + C^0, a(1) = 7 = A + B + C.

Links

Crossrefs

Cf. A141221, A141384.

Programs

  • Magma
    I:=[3,7,31]; [n le 3 select I[n] else 7*Self(n-1)-9*Self(n-2)+Self(n-3): n in [1..30]]; // Vincenzo Librandi, Oct 21 2012
  • Maple
    m:=30; S:=series( (3-14*x+9*x^2)/(1-7*x+9*x^2-x^3), x, m+1):
seq(coeff(S, x, j), j=0..m); # G. C. Greubel, Mar 30 2021
  • Mathematica
    LinearRecurrence[{7,-9,1},{3,7,31},40] (* Harvey P. Dale, May 25 2011 *)
CoefficientList[Series[(3 -14x +9x^2)/(1 -7x +9x^2 -x^3), {x, 0, 50}], x] (* Vincenzo Librandi, Oct 21 2012 *)
  • PARI
    a(n)=([0,1,0; 0,0,1; 1,-9,7]^n*[3;7;31])[1,1] \\ Charles R Greathouse IV, Feb 10 2017
  • Sage
    def A141385_list(prec):
    P. = PowerSeriesRing(ZZ, prec)
    return P( (3-14*x+9*x^2)/(1-7*x+9*x^2-x^3) ).list()
A141385_list(40) # G. C. Greubel, Mar 30 2021

Formula

G.f.: (3 - 14*x + 9*x^2)/(1 - 7*x + 9*x^2 - x^3).
a(n+3) = 7*a(n+2) - 9*a(n+1) + a(n).
a(n) = A^n + B^n + C^n, where, putting u = atan(sqrt(5319)/73), we have:
A = 5.3538557854308282... = (7 + 2*sqrt(22)*cos(u/3))/3,
B = 1.5235479602692093... = (7 - sqrt(22)*cos(u/3) + sqrt(66)*sin(u/3))/3,
C = 0.1225962542999624... = (7 - sqrt(22)*cos(u/3) - sqrt(66)*sin(u/3))/3.

Extensions

New definition by Bruno Berselli, Oct 22 2012
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