A069009 Let M denote the 6 X 6 matrix with rows / 1,1,1,1,1,1 / 1,1,1,1,1,0 / 1,1,1,1,0,0 / 1,1,1,0,0,0 / 1,1,0,0,0,0 / 1,0,0,0,0,0 / and A(n) the vector (x(n),y(n),z(n),t(n),u(n),v(n)) = M^n*A where A is the vector (1,1,1,1,1,1); then a(n) = t(n).

1, 3, 15, 59, 250, 1030, 4283, 17752, 73658, 305513, 1267344, 5257031, 21806850, 90457205, 375227042, 1556484658, 6456477531, 26782210229, 111095686086, 460837670465, 1911607611040, 7929568022610, 32892759309540

Offset

0,2

Comments

This sequence is related to the tridecagon or triskaidecagon (13-gon).

The lengths of the diagonals of the regular tridecagon are r[k] = sin(k*Pi/13)/sin(Pi/13), 1 <= k <= 6, where r[1] = 1 is the length of the edge.

Links

Table of n, a(n) for n=0..22.

P. Steinbach, Golden fields: a case for the heptagon, Math. Mag. 70 (1997), no. 1, 22-31.

Index entries for linear recurrences with constant coefficients, signature (3, 6, -4, -5, 1, 1).

Formula

G.f.: 1/(1-3*x-6*x^2+4*x^3+5*x^4-x^5-x^6). - Roger L. Bagula and Gary W. Adamson, Sep 19 2006

a(n-2) = T(n,3) with T(n,k) = sum(T(n-1,k1), k1=7-k..6), T(1,1) = T(1,2) = T(1,3) = T(1,4) = T(1,5) = 0 and T(1,6) = 1, n>=1 and 1 <= k <= 6. [Steinbach]

sum(T(n,k)*r[k], k=1..6) = r[6]^n, n>=1, with r[k] = sin(k*Pi/13)/sin(Pi/13). [Steinbach]

Maple

nmax:=22: with(LinearAlgebra): M:=Matrix([[1, 1, 1, 1, 1, 1], [1, 1, 1, 1, 1, 0], [1, 1, 1, 1, 0, 0], [1, 1, 1, 0, 0, 0], [1, 1, 0, 0, 0, 0], [1, 0, 0, 0, 0, 0]]): v:= Vector[row]([1, 1, 1, 1, 1, 1]): for n from 0 to nmax do b:=evalm(v&*M^n); a(n):=b[4] od: seq(a(n), n=0..nmax); # Johannes W. Meijer, Aug 03 2011
nmax:=24: m:=6: for k from 1 to m-1 do T(1, k):=0 od: T(1, m):=1: for n from 2 to nmax do for k from 1 to m do T(n, k):= add(T(n-1, k1), k1=m-k+1..m) od: od: for n from 1 to nmax/3 do seq(T(n, k), k=1..m) od; for n from 2 to nmax do a(n-2):=T(n, 3) od: seq(a(n), n=0..nmax-2); # Johannes W. Meijer, Aug 03 2011

Mathematica

b = {1, -3, -6, 4, 5, -1, -1}; p[x_] := Sum[x^(n - 1)*b[[8 - n]], {n, 1, 7}] q[x_] := ExpandAll[x^6*p[1/x]] Table[ SeriesCoefficient[ Series[x/q[x], {x, 0, 30}], n], {n, 0, 30}] (* Roger L. Bagula and Gary W. Adamson, Sep 19 2006 *)
CoefficientList[Series[1/(1 - 3 x - 6 x^2 + 4 x^3 + 5 x^4 - x^5 - x^6), {x, 0, 33}], x] (* Vincenzo Librandi, Sep 19 2015 *)

Prog

(PARI) Vec(1/(1-3*x-6*x^2+4*x^3+5*x^4-x^5-x^6)+O(x^33)) \\ Joerg Arndt, Sep 19 2015

Crossrefs

Cf. A066170.

Cf. A006359, A069007, A069008, A069009, A070778, A006359 (offset), for x(n), y(n), z(n), t(n), u(n), v(n).

Cf. A120747 (m = 5: hendecagon or 11-gon)

Sequence in context: A309564 A062473 A218200 * A328104 A179604 A128237

Adjacent sequences: A069006 A069007 A069008 * A069010 A069011 A069012

Keyword

easy,nonn

Author

Benoit Cloitre, Apr 02 2002

Extensions

Edited by Henry Bottomley, May 06 2002

Information added by Johannes W. Meijer, Aug 03 2011

Status

approved