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

0,3

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

Table of n, a(n) for n=0..19.

John Elias, Illustration: Union of Cantor Square and Koch Snowflake fractals

Rigoberto Flórez, Robinson A. Higuita and Antara 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).

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)

From Peter Bala, Sep 22 2025: (Start)

Sum_{n >= 1} a(n)*x^(2*n)/(2*n)! = (2/5) * sinh(5*x/2)*sinh(x/2).

Sum_{n >= 1} a(n)/(2*n)! = (1/10) * (e^3 - e^2 - 1/e^2 + 1/e^3). (End)

Mathematica

a[n_]:=(9^n-4^n)/5; a[Range[0, 60]] (* Vladimir Joseph Stephan Orlovsky, Feb 01 2011 *)
LinearRecurrence[{13, -36}, {0, 1}, 30] (* Harvey P. Dale, Jan 16 2026 *)

Prog

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

Crossrefs

Cf. A015441, A002450.

Sequence in context: A395828 A081042 A329019 * A187732 A031138 A360190

Adjacent sequences: A016150 A016151 A016152 * A016154 A016155 A016156

Keyword

nonn,easy

Author

N. J. A. Sloane

Status

approved