OFFSET
-20,1
COMMENTS
The recurrence relation a(n+1) = a(n) + a(n-1) defines the Fibonacci sequence for all (positive and negative) integer indices, given any two values with indices of opposite parity, e.g., a(0) and a(1), or a(-1) and a(42). Any other Fibonacci-type sequence {b(n)} satisfying this recurrence relation can be written as b(n) = b(1)*A000045(n) + b(0)*A000045(n-1). This can be seen from the fact that the set of all sequences satisfying a given linear recurrence relation of order 2 with constant coefficients forms a vector space of dimension two. So each element (sequence) of this space is a linear combination of any two elements which are not proportional to each other and thus form a base. The most natural choice of such a base could be the two sequences having (b(0), b(1)) = (0, 1) resp (1, 0). These are A000045 and n -> A000045(n-1) = A212804 (extended to negative indices, if needed). - M. F. Hasler, May 10 2017
LINKS
G. C. Greubel, Table of n, a(n) for n = -20..1000
Philipp Fahr and Claus Michael Ringel, Categorification of the Fibonacci Numbers Using Representations of Quivers
Index entries for linear recurrences with constant coefficients, signature (1,1).
FORMULA
a(n) = a(n-1) + a(n-2). - R. J. Mathar, Nov 30 2008
G.f.: (-6765 + 10946*x)/((1-x-x^2)*x^20). - G. C. Greubel, Jan 10 2020
MAPLE
with(combinat):seq(fibonacci(n), n=-20..30); # G. C. Greubel, Jan 10 2020
MATHEMATICA
Array[Fibonacci, 51, -20] (* Michael De Vlieger, May 10 2017 *)
Fibonacci[Range[-20, 30]] (* G. C. Greubel, Jan 10 2020 *)
PROG
(PARI) a(n)=fibonacci(n) \\ M. F. Hasler, May 10 2017
(Magma) [Fibonacci(n): n in [-20..30]]; // G. C. Greubel, Jan 10 2020
(Sage) [fibonacci(n) for n in (-20..50)] # G. C. Greubel, Jan 10 2020
(GAP) List([-20..30], n-> Fibonacci(n)); # G. C. Greubel, Jan 10 2020
CROSSREFS
KEYWORD
sign,easy,less
AUTHOR
Roger L. Bagula, Nov 05 2008
EXTENSIONS
Extended to n = -20 .. 30 by M. F. Hasler, May 10 2017
STATUS
approved