A060934 Second column of Lucas bisection triangle (even part).

1, 17, 80, 303, 1039, 3364, 10493, 31885, 95032, 279051, 809771, 2327372, 6636025, 18794633, 52925984, 148303719, 413768263, 1150029940, 3185625077, 8797726981, 24230897416, 66574108227

Offset: 0

Author

Wolfdieter Lang, Apr 20 2001

Keywords

• nonn
• easy

Comments

Numerator of g.f. is row polynomial Sum_{m=0..3} A061186(2, m)*x^m.

Links

• G. C. Greubel, Table of n, a(n) for n = 0..1000
• É. Czabarka, R. Flórez, and 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, A001871, A001906, A002878, A060923, A061186.

Programs

Magma

[2*n*Lucas(2*n+2) + Fibonacci(2*n+2): n in [0..30]]; // G. C. Greubel, Apr 09 2021

Mathematica

LinearRecurrence[{6,-11,6,-1}, {1,17,80,303}, 31] (* G. C. Greubel, Apr 09 2021 *)
CoefficientList[Series[(1+11x-11x^2+4x^3)/(1-3x+x^2)^2,{x,0,30}],x] (* Harvey P. Dale, Aug 28 2021 *)

Sage

[2*n*lucas_number2(2*n+2,1,-1) + fibonacci(2*n+2) for n in (0..30)] # G. C. Greubel, Apr 09 2021

Formula

a(n) = A060923(n+1, 1).

G.f.: (1 + 11*x - 11*x^2 + 4*x^3)/(1 - 3*x + x^2)^2.

a(n) = 2*n*Lucas(2*n+2) + Fibonacci(2*n+2). - G. C. Greubel, Apr 09 2021