

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
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OFFSET

20,1


COMMENTS

The recurrence relation a(n+1) = a(n) + a(n1) 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 Fibonaccitype sequence {b(n)} satisfying this recurrence relation can be written as b(n) = b(1)*A000045(n) + b(0)*A000045(n1). 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(n1) = 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(n1) + a(n2).  R. J. Mathar, Nov 30 2008
G.f.: (6765 + 10946*x)/((1xx^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



