

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
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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(n2)*rho(9) + b(n1)*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 9gon. 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 entries for 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/(13x+x^3). a(n) = 3*a(n1)a(n3), n>0.
a(n)=sum{k=0..floor(n/3), binomial(n2k, k)(1)^k*3^(n3k)}.  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


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
G.f. = 1 + 3*x + 9*x^2 + 26*x^3 + 75*x^4 + 216*x^5 + 622*x^6 + 1791*x^7 + ...


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



