|
|
A099843
|
|
A transform of the Fibonacci numbers.
|
|
2
|
|
|
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
(list;
graph;
refs;
listen;
history;
text;
internal format)
|
|
|
OFFSET
|
0,2
|
|
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
|
|
|
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).
|
|
MAPLE
|
a:= n-> (<<0|1>, <1|-4>>^n.<<1, -5>>)[1, 1]:
|
|
MATHEMATICA
|
LinearRecurrence[{-4, 1}, {1, -5}, 30] (* Harvey P. Dale, Aug 13 2015 *)
|
|
PROG
|
(Magma) [(-1)^n*Fibonacci(3*n+2): n in [0..40]]; // G. C. Greubel, Apr 20 2023
(SageMath) [(-1)^n*fibonacci(3*n+2) for n in range(41)] # G. C. Greubel, Apr 20 2023
|
|
CROSSREFS
|
Cf. A084326 (shifted unsigned inverse binomial transform), A152174 (binomial transform).
|
|
KEYWORD
|
easy,sign
|
|
AUTHOR
|
|
|
STATUS
|
approved
|
|
|
|