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A091024
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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).
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4
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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
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OFFSET
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0,3
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COMMENTS
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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)
(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 (see cf. A187503, A187504, A187505, A187506). -L. Edson Jeffery, Mar 15 2011. (End)
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REFERENCES
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Jay Kappraff, "Beyond Measure, A Guided Tour Through Nature, Myth and Number" (p.497 gives the analogous case for the Heptagon).
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LINKS
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Table of n, a(n) for n=0..26.
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FORMULA
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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.
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EXAMPLE
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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
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MATHEMATICA
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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}] (from Robert G. Wilson v Feb 21 2005)
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CROSSREFS
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Cf. A006357, A076264.
Sequence in context: A145519 A030224 A114624 * A151430 A083309 A164979
Adjacent sequences: A091021 A091022 A091023 * A091025 A091026 A091027
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KEYWORD
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nonn
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AUTHOR
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Gary W. Adamson, Dec 14 2003
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EXTENSIONS
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More terms from Robert G. Wilson v, Feb 21 2005
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STATUS
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approved
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