A038213 Top line of 3-wave sequence A038196, also bisection of A006356.
1, 3, 14, 70, 353, 1782, 8997, 45425, 229347, 1157954, 5846414, 29518061, 149034250, 752461609, 3799116465, 19181424995, 96845429254, 488964567014, 2468741680809, 12464472679038, 62932092237197, 317738931708801
Offset: 0
Examples
G.f. = 1 + 3*x + 14*x^2 + 70*x^3 + 353*x^4 + 1782*x^5 + 8997*x^6 + 45425*x^7 + ...
Links
- Johann Cigler, Number of bounded Dyck paths with "negative length", MathOverflow question, Sep 26 2020.
- S. Morier-Genoud, V. Ovsienko and S. Tabachnikov, 2-frieze patterns and the cluster structure of the space of polygons, Annales de l'institut Fourier, 62 no. 3 (2012), 937-987; arXiv:1008.3359 [math.AG], 2010-2011. - _N. J. A. Sloane_, Dec 26 2012
- F. v. Lamoen, Wave sequences
- Index entries for linear recurrences with constant coefficients, signature (6, -5, 1).
Crossrefs
Cf. A080937.
Programs
-
PARI
k=3; M(k)=matrix(k,k,i,j,min(i,j)); v(k)=vector(k,i,1); a(n)=vecmin(v(k)*M(k)^n)
-
PARI
{a(n) = if( n<0, n = -n; polcoeff( (1 - 4*x + 3*x^2) / (1 - 5*x + 6*x^2 - x^3) + x * O(x^n), n), polcoeff( (1 - 3*x + x^2) / (1 - 6*x + 5*x^2 - x^3) + x * O(x^n), n))}; /* Michael Somos, May 04 2012 */
Formula
Let v(3)=(1, 1, 1), let M(3) be the 3 X 3 matrix m(i, j) =min(i, j); then a(n)= min ( v(3)*M(3)^n). - Benoit Cloitre, Oct 03 2002
G.f.: -((1 + (-3 + q)*q)/(-1 + (-3 + q)*(-2 + q)*q)). - Wouter Meeussen, Mar 19 2005
G.f.: (1 - 3*x + x^2) / (1 - 6*x + 5*x^2 - x^3).
a(-n) = A080937(n) for all n in Z. a(n + 2) * a(n) - a(n + 1)^2 = a(-3 - n) for all n in Z. - Michael Somos, May 04 2012
Extensions
More terms from Benoit Cloitre, Oct 03 2002