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.
%I A069009 #31 Sep 12 2024 09:10:00
%S A069009 1,3,15,59,250,1030,4283,17752,73658,305513,1267344,5257031,21806850,
%T A069009 90457205,375227042,1556484658,6456477531,26782210229,111095686086,
%U A069009 460837670465,1911607611040,7929568022610,32892759309540
%N 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).
%C A069009 This sequence is related to the tridecagon or triskaidecagon (13-gon).
%C A069009 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.
%H A069009 P. Steinbach, <a href="http://www.jstor.org/stable/2691048">Golden fields: a case for the heptagon</a>, Math. Mag. 70 (1997), no. 1, 22-31.
%H A069009 <a href="/index/Rec#order_06">Index entries for linear recurrences with constant coefficients</a>, signature (3, 6, -4, -5, 1, 1).
%F A069009 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
%F A069009 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]
%F A069009 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]
%p A069009 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
%p A069009 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
%t A069009 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 *)
%t A069009 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 *)
%o A069009 (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
%Y A069009 Cf. A066170.
%Y A069009 Cf. A006359, A069007, A069008, A069009, A070778, A006359 (offset), for x(n), y(n), z(n), t(n), u(n), v(n).
%Y A069009 Cf. A120747 (m = 5: hendecagon or 11-gon)
%K A069009 easy,nonn
%O A069009 0,2
%A A069009 _Benoit Cloitre_, Apr 02 2002
%E A069009 Edited by _Henry Bottomley_, May 06 2002
%E A069009 Information added by _Johannes W. Meijer_, Aug 03 2011