|
| |
|
|
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).
|
|
6
| |
|
|
1, 3, 15, 59, 250, 1030, 4283, 17752, 73658, 305513, 1267344, 5257031, 21806850, 90457205, 375227042, 1556484658, 6456477531, 26782210229, 111095686086, 460837670465, 1911607611040, 7929568022610, 32892759309540
(list; graph; refs; listen; history; internal format)
|
|
|
|
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.
|
|
|
REFERENCES
| Peter Steinbach, Golden Fields: A case for the heptagon, Mathematics Magazine 70 (1997) p. 22-31.
|
|
|
FORMULA
| G.f.: 1/(1-3*x-6*x^2+4*x^3+5*x^4-x^5-x^6). - Roger Bagula and Gary 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
| From Johannes W. Meijer, Aug 03 2011: (Start)
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);
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); # (End)
|
|
|
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 Bagula and Gary Adamson, Sep 19 2006
|
|
|
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: A049178 A049150 A062473 * A179604 A176311 A036750
Adjacent sequences: A069006 A069007 A069008 * A069010 A069011 A069012
|
|
|
KEYWORD
| easy,nonn
|
|
|
AUTHOR
| Benoit Cloitre (benoit7848c(AT)orange.fr), Apr 02 2002
|
|
|
EXTENSIONS
| Edited by Henry Bottomley (se16(AT)btinternet.com), May 06 2002.
Information added by Johannes W. Meijer, Aug 03 2011
|
| |
|
|
