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A091024 Let v(0) be the column vector (1,0,0,0)'; for n>0, let v(n) = [1 1 1 1 / 1 1 1 0 / 1 1 0 0/ 1 0 0 0] v(n-1). Sequence gives third entry of v(n).

%I A091024 #21 Feb 18 2016 11:21:02
%S A091024 0,1,2,7,19,56,160,462,1329,3828,11021,31735,91376,263108,757588,
%T A091024 2181389,6281058,18085587,52075371,149945056,431749580,1243173370,
%U A091024 3579575053,10306975580,29677753369,85453685055,246054079584
%N A091024 Let v(0) be the column vector (1,0,0,0)'; for n>0, let v(n) = [1 1 1 1 / 1 1 1 0 / 1 1 0 0/ 1 0 0 0] v(n-1). Sequence gives third entry of v(n).
%C A091024 First entry of v(n) gives 1,1,4,10,30,85 = A006357 prefixed with an initial 1, the second entry gives 0,1,3,9,26,... = A076264 prefixed with an initial 0.
%C A091024 A sequence derived from 9-gonal diagonal ratios.
%C A091024 a(n)/a(n-1) converges to D = 2.879385... = longest 9-Gon diagonal with edge = 1. E.g., a(7)/a(6) = 707/246 = 2.873983...(a(n)/a(n-1) of all 4 columns converge to 2.8739...). For each row, left to right, terms converge upon 9-Gon ratios: (2.879...):(2.53208...):(1.87938...):(1) Example: row 7 = 707 622 462 246, from A006357, A076264, A091024 and A006357(offset), respectively. The ratios 707/246, 622/246, 462/246 and 246/246 are: (2.8739...):(2.528...):(1.87804...):(1)
%C A091024 From _L. Edson Jeffery_, Mar 15 2011: (Start)
%C A091024 In fact, the above ratios (2.8739...):(2.528...):(1.87804...):(1) converge to Q_3(w):Q_2(w):Q_1(w):Q_0(w), where the polynomials Q_r(w) are defined by Q_r(w)=w*Q_(r-1)(w)-Q_(r-2)(w) (r>1), Q_0(w)=1, Q_1(w)=w, and w=2*cos(Pi/9).
%C A091024 Moreover, this sequence and a variant of its g.f. are related to rhombus substitution tilings showing 9-fold rotational symmetry (cf. A187503, A187504, A187505, A187506). (End)
%D A091024 Jay Kappraff, "Beyond Measure, A Guided Tour Through Nature, Myth and Number" (p. 497 gives the analogous case for the Heptagon).
%H A091024 Harvey P. Dale, <a href="/A091024/b091024.txt">Table of n, a(n) for n = 0..1000</a>
%H A091024 <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (2,3,-1,-1).
%F A091024 Recurrence: a(n) = 2*a(n-1) + 3*a(n-2) - a(n-3) - a(n-4), with initial conditions {a(k)}={0,1,2,7}, k=0,1,2,3. - _L. Edson Jeffery_, Mar 15 2011
%F A091024 G.f.: x/(1 - 2*x - 3*x^2 + x^3 + x^4). - _L. Edson Jeffery_, Mar 15 2011
%F A091024 G.f.: Q(0)*x/(2+2*x) , where Q(k) = 1 + 1/(1 - x*(12*k-3 + x^2)/( x*(12*k+3 + x^2 ) - 1/Q(k+1) )); (continued fraction). - _Sergei N. Gladkovskii_, Sep 12 2013
%e A091024 A006357, A076264, a(n) and A006357 (offset) gives the 4 components of v(n) transposed:
%e A091024 1 0 0 0
%e A091024 1 1 1 1
%e A091024 4 3 2 1
%e A091024 10 9 7 4
%e A091024 30 26 19 10
%e A091024 85 75 56 30
%t A091024 a[n_] := (MatrixPower[{{1, 1, 1, 1}, {1, 1, 1, 0}, {1, 1, 0, 0}, {1, 0, 0, 0}}, n].{{1}, {0}, {0}, {0}})[[3, 1]]; Table[ a[n], {n, 0, 26}] (* _Robert G. Wilson v_, Feb 21 2005 *)
%t A091024 LinearRecurrence[{2,3,-1,-1},{0,1,2,7},30] (* _Harvey P. Dale_, Feb 18 2016 *)
%Y A091024 Cf. A006357, A076264.
%K A091024 nonn
%O A091024 0,3
%A A091024 _Gary W. Adamson_, Dec 14 2003
%E A091024 More terms from _Robert G. Wilson v_, Feb 21 2005