A016153 a(n) = (9^n-4^n)/5.

0, 1, 13, 133, 1261, 11605, 105469, 953317, 8596237, 77431669, 697147165, 6275373061, 56482551853, 508359743893, 4575304803901, 41178011670565, 370603178776909, 3335432903959477, 30018913315504477, 270170288559017029

Offset: 0

Author

N. J. A. Sloane

Keywords

• nonn
• easy

Comments

a(n) is also the coefficient of x^(n-1) in the bivariate Fibonacci polynomials F(n)(x,y)=xF(n-1)(x,y)+yF(n-2)(x,y),F(0)(x,y)=0,F(1)(x,y)=1, when we write 13x for x and -36x^2 for y. - Mario Catalani (mario.catalani(AT)unito.it), Dec 09 2002

Links

• John Elias, Illustration: Union of Cantor Square and Koch Snowflake fractals
• R. Flórez, R. A. Higuita and A. Mukherjee, Alternating Sums in the Hosoya Polynomial Triangle, Article 14.9.5 Journal of Integer Sequences, Vol. 17 (2014).
• Index entries for linear recurrences with constant coefficients, signature (13,-36).

Crossrefs

Cf. A015441, A002450.

Programs

Mathematica

Join[{a=0,b=1},Table[c=13*b-36*a;a=b;b=c,{n,60}]](*and/or*)f[n_]:=(9^n-4^n)/5;f[Range[0,60]] (* Vladimir Joseph Stephan Orlovsky, Feb 01 2011 *)

PARI

a(n)=(9^n-4^n)/5

Formula

G.f.: x/((1-4*x)*(1-9*x)). a(n)=13*a(n-1)-36*a(n-2).

a(n) = A015441(2*n).

From Peter Bala, Jul 23 2025: (Start)

The following products telescope. Cf. A002450:

Product_{k >= 1} 1 + 6^k/a(k+1) = 3; Product_{k >= 1} 1 - 6^k/a(k+1) = 3/13.

Product_{k >= 1} 1 + (-6)^k/a(k+1) = 3/5; Product_{k >= 1} 1 - (-6)^k/a(k+1) = 15/13.

(End)