A190960 a(n) = 3*a(n-1) - 6*a(n-2), with a(0)=0, a(1)=1.

0, 1, 3, 3, -9, -45, -81, 27, 567, 1539, 1215, -5589, -24057, -38637, 28431, 317115, 780759, 439587, -3365793, -12734901, -18009945, 22379571, 175198383, 391317723, 122762871, -1979617725, -6675430401, -8148584853, 15606827847, 95711992659, 193495010895

Offset: 0

Author

Vladimir Joseph Stephan Orlovsky, May 24 2011

Keywords

• sign
• easy

Links

• G. C. Greubel, Table of n, a(n) for n = 0..1000
• Taras Goy and Mark Shattuck, Determinants of Toeplitz-Hessenberg Matrices with Generalized Leonardo Number Entries, Ann. Math. Silesianae (2023). See p. 16.
• Index entries for linear recurrences with constant coefficients, signature (3,-6).

Crossrefs

Cf. A190958 (index to generalized Fibonacci sequences).

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 25 2018

Mathematica

LinearRecurrence[{3,-6}, {0,1}, 50]

PARI

x='x+O('x^30); concat([0], Vec(x/(1-3*x+6*x^2))) \\ G. C. Greubel, Jan 25 2018

Formula

G.f.: x/(1-3*x+6*x^2). - Philippe Deléham, Oct 11 2011

a(n) = (i/sqrt(15))*((3/2 - i*sqrt(15)/2)^n - (3/2 + i*sqrt(15)/2)^n), where i=sqrt(-1). - Taras Goy, Jan 04 2025

E.g.f.: 2*exp(3*x/2)*sin(sqrt(15)*x/2)/sqrt(15). - Stefano Spezia, Jan 05 2025