A061171 One half of second column of Lucas bisection triangle (odd part).

3, 19, 79, 283, 940, 2982, 9171, 27581, 81557, 237995, 687158, 1966764, 5588259, 15780103, 44323195, 123920827, 345062176, 957403026, 2647935987, 7302634865, 20087869313, 55128445259, 150971982314

Offset: 0

Author

Wolfdieter Lang, Apr 20 2001

Keywords

• nonn
• easy

Comments

Numerator of g.f. is on half of row polynomial Sum_{m=0..2} A061187(1,m) * x^m.

Links

• G. C. Greubel, Table of n, a(n) for n = 0..1000
• É. Czabarka, R. Flórez, L. Junes, A Discrete Convolution on the Generalized Hosoya Triangle, Journal of Integer Sequences, 18 (2015), #15.1.6.
• Index entries for linear recurrences with constant coefficients, signature (6,-11,6,-1).

Crossrefs

Cf. A000032, A000045, A001906, A002878, A005248.

Programs

Magma

I:=[3,19,79,283]; [n le 4 select I[n] else 6*Self(n-1) - 11*Self(n-2) + 6*Self(n-3) - Self(n-4): n in [1..30]]; // G. C. Greubel, Dec 21 2017

Mathematica

CoefficientList[Series[(1+x)(3-2x)/(1-3x+x^2)^2,{x,0,30}],x] (* or *) LinearRecurrence[{6,-11,6,-1},{3,19,79,283},30] (* Harvey P. Dale, Oct 11 2012 *)

PARI

my(x='x+O('x^30)); Vec((1+x)*(3-2*x)/(1-3*x+x^2)^2) \\ G. C. Greubel, Dec 21 2017

Formula

2*a(n) = A060924(n+1, 1).

G.f.: (1+x)*(3-2*x)/(1-3*x+x^2)^2.

a(n) = 6*a(n-1) - 11*a(n-2) + 6*a(n-3) - a(n-4), with a(0)=3, a(1)=19, a(2)=79, a(3)=283. - Harvey P. Dale, Oct 11 2012

a(n) = Fibonacci(2*n+4) + n*Lucas(2*n+3). - Lechoslaw Ratajczak, May 06 2020