A076264 Number of ternary (0,1,2) sequences without a consecutive '012'.

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

Offset: 0

Author

John L. Drost, Nov 05 2002

Keywords

• nonn
• easy

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 -0.6527036445, and they are therefore irrelevant for this sequence. - Wolfdieter Lang, Nov 08 2013

a(n) is also the number of ternary (0,1,2) sequences of length n without a consecutive '110' because the patterns A=012 and B=110 have the same autocorrelation, i.e., AA=100=BB, in the sense of Guibas and Odlysko (1981). (A cyclic version of this sequence can be found in sequence A274018.) - Petros Hadjicostas, Sep 12 2017

Examples

			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 + ...
		

References

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

Links

• Harvey P. Dale, Table of n, a(n) for n = 0..1000
• Taras Goy and Mark Shattuck, Toeplitz-Hessenberg determinant formulas for the sequence F_n-1, Online J. Anal. Comb. (2025) Vol. 19, Paper 1, 1-26. See p. 12.
• Leo J. Guibas and Andrew M. Odlyzko, String overlaps, pattern matching, and nontransitive games, J. Combin. Theory Ser. A 30 (1981), 183-208. [Comment: The authors use generating functions in terms of z^{-1}. To get the g.f. of the sequence, as shown below in the FORMULA section, let x=z^{-1} and perform simple algebra. There are some minor typos in Theorem 2.1, p. 191, that can be easily corrected by looking at the proof. - _Petros Hadjicostas_, Sep 12 2017]
• Yun-Tak Oh, Hosho Katsura, Hyun-Yong Lee, and Jung Hoon Han, Proposal of a spin-one chain model with competing dimer and trimer interactions, arXiv:1709.01344 [cond-mat.str-el], 2017.
• Index entries for linear recurrences with constant coefficients, signature (3,0,-1).

Crossrefs

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

Programs

GAP

List([0..25],n->Sum([0..Int(n/3)],k->Binomial(n-2*k,k)*(-1)^k*3^(n-3*k))); # Muniru A Asiru, Feb 20 2018

Mathematica

LinearRecurrence[{3,0,-1},{1,3,9},30] (* Harvey P. Dale, Feb 28 2016 *)

PARI

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

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 3 X 3 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