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

0,3

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)

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).

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

Example

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

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 *)

Crossrefs

Cf. A006357, A076264.

Sequence in context: A145519 A030224 A114624 * A275289 A151430 A083309

Adjacent sequences: A091021 A091022 A091023 * A091025 A091026 A091027

Keyword

nonn

Author

Gary W. Adamson, Dec 14 2003

Extensions

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

Status

approved