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A147316 Fibonacci numbers (A000045) starting at offset -20. 3
-6765, 4181, -2584, 1597, -987, 610, -377, 233, -144, 89, -55, 34, -21, 13, -8, 5, -3, 2, -1, 1, 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, 610, 987, 1597, 2584, 4181, 6765, 10946, 17711, 28657, 46368, 75025, 121393, 196418, 317811, 514229, 832040 (list; graph; refs; listen; history; text; internal format)
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

Cf. A000285, A022113, A001060.

Of course A000045 is the main entry for the Fibonacci numbers.

See also A039834 for A000045 extended to negative indices.

Sequence in context: A204799 A151637 A031986 * A206096 A238911 A031832

Adjacent sequences:  A147313 A147314 A147315 * A147317 A147318 A147319

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

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Last modified September 24 17:09 EDT 2020. Contains 337321 sequences. (Running on oeis4.)