A100545 Expansion of (7-2*x) / (1-3*x+x^2).

7, 19, 50, 131, 343, 898, 2351, 6155, 16114, 42187, 110447, 289154, 757015, 1981891, 5188658, 13584083, 35563591, 93106690, 243756479, 638162747, 1670731762, 4374032539, 11451365855, 29980065026, 78488829223, 205486422643, 537970438706, 1408424893475, 3687304241719, 9653487831682, 25273159253327

Offset: 0

Author

Creighton Dement, Dec 31 2004

Keywords

• nonn
• easy

Comments

A Floretion integer sequence relating to Fibonacci numbers.

Inverse binomial transform of A013655; inversion of A097924.

Links

• Colin Barker, Table of n, a(n) for n = 0..1000
• Mark W. Coffey, James L. Hindmarsh, Matthew C. Lettington, John Pryce, On Higher Dimensional Interlacing Fibonacci Sequences, Continued Fractions and Chebyshev Polynomials, arXiv:1502.03085 [math.NT], 2015 (see p. 31).
• Tanya Khovanova, Recursive Sequences
• Index entries for linear recurrences with constant coefficients, signature (3,-1).

Crossrefs

Cf. A000032, A000045, A001906, A005248, A013655, A097924.

Programs

GAP

List([0..30], n-> Fibonacci(2*n+4) +Lucas(1,-1,2*n+3)[2] ); # G. C. Greubel, Jan 17 2020

Magma

[Fibonacci(2*n+4) +Lucas(2*n+3): n in [0..30]]; // G. C. Greubel, Jan 17 2020

Maple

F := proc(n) combinat[fibonacci](n) ; end: A100545 := proc(n) 4*F(2*(n+1)) + F(2*n+1)+F(2*n+3) ; end: for n from 0 to 30 do printf("%d,",A100545(n)) ; od ; # R. J. Mathar, Oct 26 2006

Mathematica

Table[Fibonacci[2*(n+2)] + LucasL[2*n+3], {n,0,30}] (* G. C. Greubel, Jan 17 2020 *)

PARI

Vec((7-2*x)/(1-3*x+x^2) + O(x^30)) \\ Michel Marcus, Feb 11 2015

Sage

[fibonacci(2*n+4) +lucas_number2(2*n+3,1,-1) for n in (0..30)] # G. C. Greubel, Jan 17 2020

Formula

a(n-1) = 4*Fibonacci(2*n) + Fibonacci(2*n-1) + Fibonacci(2*n+1).

a(n) + a(n+1) = A055849(n+2).

a(n) = 3*a(n-1) - a(n-2) with a(0)=7 and a(1)=19. - Philippe Deléham, Nov 16 2008

a(n) = (2^(-1-n)*((3-sqrt(5))^n*(-17+7*sqrt(5)) + (3+sqrt(5))^n*(17+7*sqrt(5)))) / sqrt(5). - Colin Barker, Oct 14 2015

From G. C. Greubel, Jan 17 2020: (Start)

a(n) = Fibonacci(2*n+4) + Lucas(2*n+3).

E.g.f.: 2*exp(3*t/2)*(cosh(sqrt(5)*t/2) + (4/sqrt(5))*sinh(sqrt(5)*t/2)). (End)

Extensions

Corrected and extended by T. D. Noe and R. J. Mathar, Oct 26 2006