A083858 Expansion of x/(1 - 3*x - 6*x^2).

0, 1, 3, 15, 63, 279, 1215, 5319, 23247, 101655, 444447, 1943271, 8496495, 37149111, 162426303, 710173575, 3105078543, 13576277079, 59359302495, 259535569959, 1134762524847, 4961500994295, 21693078131967, 94848240361671, 414703189876815, 1813199011800471

Offset: 0

Author

Paul Barry, May 06 2003

Keywords

• easy
• nonn
• changed

Comments

Binomial transform of A015443. A row of array A083857.

Pisano period lengths: 1, 1, 1, 1, 12, 1, 8, 1, 1, 12, 110, 1, 168, 8, 12, 2, 16, 1, 360, 12, ... - R. J. Mathar, Aug 10 2012

Links

• G. C. Greubel, Table of n, a(n) for n = 0..1000
• Index entries for linear recurrences with constant coefficients, signature (3,6).

Crossrefs

Cf. A015523, A015524.

Programs

Magma

I:=[0,1]; [n le 2 select I[n] else 3*Self(n-1) + 6*Self(n-2): n in [1..30]]; // G. C. Greubel, Jan 16 2018

Mathematica

a[n_]:=(MatrixPower[{{1,2},{1,-4}},n].{{1},{1}})[[2,1]]; Table[Abs[a[n]],{n,-1,40}] (* Vladimir Joseph Stephan Orlovsky, Feb 19 2010 *)
LinearRecurrence[{3,6}, {0,1}, 30] (* G. C. Greubel, Jan 16 2018 *)

PARI

x='x+O('x^30); concat([0], Vec(x/(1-3*x-6*x^2))) \\ G. C. Greubel, Jan 16 2018

Sage

[lucas_number1(n,3,-6) for n in range(0, 24)] # Zerinvary Lajos, Apr 22 2009

Formula

a(n) = 3*a(n-1) + 6*a(n-2), a(0)=0, a(1)=1.

a(n) = (3*sqrt(33)/2 + 21/2)^(n/2)/sqrt(33) - (21/2 - 3*sqrt(33)/2)^(n/2)*(-1)^n/sqrt(33).

G.f.: Q(0)*x/2, where Q(k) = 1 + 1/(1 - x*(6*k+3 + 6*x )/( x*(6*k+6 + 6*x ) + 1/Q(k+1) )); (continued fraction). - Sergei N. Gladkovskii, Sep 21 2013

a(n) = B(n, k + 2^(n-1)) - B(n,k) where B(n,k) is formed by the family of recursions b(n) = 3*(b(n-1) + b(n-2))/2, with b(0) = 1 and b(1) = k, as explained further in A249861. - Richard R. Forberg, Nov 04 2014

a(n) = Sum_{k=0..n-1} 3^k * 2^(n-1-k) * binomial(k,n-1-k). - Seiichi Manyama, Aug 31 2025