A099843 A transform of the Fibonacci numbers.

1, -5, 21, -89, 377, -1597, 6765, -28657, 121393, -514229, 2178309, -9227465, 39088169, -165580141, 701408733, -2971215073, 12586269025, -53316291173, 225851433717, -956722026041, 4052739537881, -17167680177565, 72723460248141, -308061521170129, 1304969544928657

Offset: 0

Author

Paul Barry, Oct 27 2004

Keywords

• easy
• sign

Comments

The g.f. is the transform of the g.f. of A000045 under the mapping G(x) -> (-1/(1+x))*G((x-1)/(x+1)). In general this mapping transforms x/(1-k*x-k*x^2) into (1-x)/(1 + 2(k+1)*x - (2*k-1)*x^2).

Pisano period lengths: 1, 1, 8, 2, 20, 8, 16, 4, 8, 20, 10, 8, 28, 16, 40, 8, 12, 8, 6, 20, ... - R. J. Mathar, Aug 10 2012

Links

• G. C. Greubel, Table of n, a(n) for n = 0..1000
• Tanya Khovanova, Recursive Sequences
• Index entries for linear recurrences with constant coefficients, signature (-4,1).

Crossrefs

Cf. A000045, A001076, A015448, A099842.

Cf. A084326 (shifted unsigned inverse binomial transform), A152174 (binomial transform).

Programs

Magma

[(-1)^n*Fibonacci(3*n+2): n in [0..40]]; // G. C. Greubel, Apr 20 2023

Maple

a:= n-> (<<0|1>, <1|-4>>^n.<<1, -5>>)[1,1]:
seq(a(n), n=0..24);  # Alois P. Heinz, Apr 21 2023

Mathematica

CoefficientList[Series[(1-x)/(1+4*x-x^2), {x,0,30}], x] (* Vladimir Joseph Stephan Orlovsky, Jun 10 2011 *)
LinearRecurrence[{-4,1},{1,-5},30] (* Harvey P. Dale, Aug 13 2015 *)

SageMath

[(-1)^n*fibonacci(3*n+2) for n in range(41)] # G. C. Greubel, Apr 20 2023

Formula

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

a(n) = (sqrt(5)-2)^n * (1/2 - 3*sqrt(5)/10) + (-sqrt(5)-2)^n * (1/2 + 3*sqrt(5)/10).

a(n) = (-1)^n*Fibonacci(3*n+2).

a(n) = -4*a(n-1) + a(n-2), a(0)=1, a(1)=-5. - Philippe Deléham, Nov 03 2008

a(n) = (-1)^n*(A001076(n) + A001076(n+1)). - R. J. Mathar, Aug 10 2012

a(n) = (-1)^n*A015448(n+1). - R. J. Mathar, May 07 2019