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A076264 Number of ternary (0,1,2) sequences without a consecutive '012'. 22
1, 3, 9, 26, 75, 216, 622, 1791, 5157, 14849, 42756, 123111, 354484, 1020696, 2938977, 8462447, 24366645, 70160958, 202020427, 581694636, 1674922950, 4822748423, 13886550633, 39984728949, 115131438424, 331507764639 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,2

COMMENTS

A transform of A000244 under the mapping g(x)->(1/(1+x^3))g(x/(1+x^3)). - Paul Barry, Oct 20 2004

b(n) := (-1)^n*a(n) appears in the formula for the nonpositive powers of rho(9) := 2*cos(Pi/9), when written in the power basis of the algebraic number field Q(rho(9)) of degree 3. See A187360 for the minimal polynomial C(9, x) of rho(9), and a link to the Q(2*cos(pi/n)) paper. 1/rho(9) = -3*1 + 0*rho(9) + 1*rho(9)^2 (see A230079, row n=5). 1/rho(9)^n = b(n)*1 + b(n-2)*rho(9) + b(n-1)*rho(9)^2, n>=0, with b(-1) = 0 = b(-2). - Wolfdieter Lang, Nov 04 2013

The limit b(n+1)/b(n) = -a(n+1)/a(n) for n -> infinity is -tau(9):= -(1 + rho(9)) = 1/(2*cos(Pi*5/9)), approximately -2.445622407. tau(9) is known to be the length ratio (longest diagonal)/side in the regular 9-gon. This limit follows from the b(n)-recurrence and the solutions of X^3 + 3*X^2 - 1 = 0, which are given by the inverse of the known solutions of the minimal polynomial C(9, x) of rho(9) (see A187360). The other two X solutions are 1/rho(9) = -3+rho(9)^2, approximately 0.5320888860 and 1/(2*cos(Pi*7/9)) = 1+rho(9)-rho(9)^2, approximately -.6527036445, and they are therefore irrelevant for this sequence. - Wolfdieter Lang, Nov 08 2013

REFERENCES

A. Tucker, Applied Combinatorics, 4th ed. p. 277

LINKS

Table of n, a(n) for n=0..25.

Index to sequences with linear recurrences with constant coefficients, signature (3,0,-1).

FORMULA

a(n) is asymptotic to g*c^n where c=cos(Pi/18)/cos(7*Pi/18) and g is the largest real root of : 81*x^3 - 81*x^2 - 9*x + 1 = 0. - Benoit Cloitre, Nov 06 2002

G.f.: 1/(1-3x+x^3). a(n) = 3*a(n-1)-a(n-3), n>0.

a(n)=sum{k=0..floor(n/3), binomial(n-2k, k)(-1)^k*3^(n-3k)}. - Paul Barry, Oct 20 2004

a(n) = middle term in M^(n+1) * [1 0 0], where M = the 3X3 matrix [2 1 1 / 1 1 0 / 1 0 0]. Right term = A052536(n), left term = A052536(n+1). - Gary W. Adamson, Sep 05 2005

G.f.: Q(0)/2 , 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

1/rho(9)^3 = -26*1 - 3*rho(9) + 9*rho(9)^2, (approximately 0.15064426) with rho(9) given in the Nov 04 2013 comment above. - Wolfdieter Lang, Nov 04 2013

PROG

(PARI) a(n)=if(n<0, 0, polcoeff(1/(1-3*x+x^3)+x*O(x^n), n))

CROSSREFS

The g.f. corresponds to row 3 of triangle A225682.

Sequence in context: A077845 A171277 A000243 * A018919 A123941 A005774

Adjacent sequences:  A076261 A076262 A076263 * A076265 A076266 A076267

KEYWORD

nonn,easy

AUTHOR

John L. Drost, Nov 05 2002

STATUS

approved

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Last modified October 25 20:39 EDT 2014. Contains 248557 sequences.