A190959 a(n) = 3*a(n-1) - 5*a(n-2), with a(0)=0, a(1)=1.

0, 1, 3, 4, -3, -29, -72, -71, 147, 796, 1653, 979, -5328, -20879, -35997, -3596, 169197, 525571, 730728, -435671, -4960653, -12703604, -13307547, 23595379, 137323872, 293994721, 195364803, -883879196, -3628461603, -6465988829, -1255658472, 28562968729

Offset: 0

Author

Vladimir Joseph Stephan Orlovsky, May 24 2011

Keywords

• sign
• easy

Comments

This is the Lucas U(P=3, Q=5) sequence. - R. J. Mathar, Oct 24 2012

a(n+2)/a(n+1) equals the continued fraction 3 - 5/(3 - 5/(3 - 5/(3 - ... - 5/3))) with n 5's. - Greg Dresden, Oct 06 2019

Links

• G. C. Greubel, Table of n, a(n) for n = 0..1000
• Wikipedia, Lucas sequence
• Index entries for linear recurrences with constant coefficients, signature (3,-5).

Crossrefs

Cf. A190958 (index to generalized Fibonacci sequences), A190970 (binomial transf.), A106852 (inv. bin. transf., shifted).

Programs

Magma

I:=[0,1]; [n le 2 select I[n] else 3*Self(n-1) - 5*Self(n-2): n in [1..30]]; // G. C. Greubel, Jan 25 2018

Mathematica

LinearRecurrence[{3,-5}, {0,1}, 50]

PARI

x='x+O('x^30); concat([0], Vec(x/(1-3x+5*x^2))) \\ G. C. Greubel, Jan 25 2018

Formula

G.f.: x/(1 - 3*x + 5*x^2). - Philippe Deléham, Oct 11 2011

E.g.f.: 2*exp(3*x/2)*sin(sqrt(11)*x/2)/sqrt(11). - Stefano Spezia, Oct 06 2019