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).

0, 1, 2, 7, 19, 56, 160, 462, 1329, 3828, 11021, 31735, 91376, 263108, 757588, 2181389, 6281058, 18085587, 52075371, 149945056, 431749580, 1243173370, 3579575053, 10306975580, 29677753369, 85453685055, 246054079584

Offset: 0

Author

Gary W. Adamson, Dec 14 2003

Keywords

• nonn

Comments

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.

A sequence derived from 9-gonal diagonal ratios.

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)

From L. Edson Jeffery, Mar 15 2011: (Start)

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).

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)

Examples

			A006357, A076264, a(n) and A006357 (offset) gives the 4 components of v(n) transposed:
1 0 0 0
1 1 1 1
4 3 2 1
10 9 7 4
30 26 19 10
85 75 56 30
		

References

• Jay Kappraff, "Beyond Measure, A Guided Tour Through Nature, Myth and Number" (p. 497 gives the analogous case for the Heptagon).

Links

• Harvey P. Dale, Table of n, a(n) for n = 0..1000
• Index entries for linear recurrences with constant coefficients, signature (2,3,-1,-1).

Crossrefs

Cf. A006357, A076264.

Programs

Mathematica

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 *)
LinearRecurrence[{2,3,-1,-1},{0,1,2,7},30] (* Harvey P. Dale, Feb 18 2016 *)

Formula

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

G.f.: x/(1 - 2*x - 3*x^2 + x^3 + x^4). - L. Edson Jeffery, Mar 15 2011

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

Extensions

More terms from Robert G. Wilson v, Feb 21 2005