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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). 5
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 (list; graph; refs; listen; history; text; internal format)
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

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Last modified April 26 11:40 EDT 2017. Contains 285444 sequences.