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Fibonacci numbers (A000045) starting at offset -20.
4

%I #42 Sep 08 2022 08:45:38

%S -6765,4181,-2584,1597,-987,610,-377,233,-144,89,-55,34,-21,13,-8,5,

%T -3,2,-1,1,0,1,1,2,3,5,8,13,21,34,55,89,144,233,377,610,987,1597,2584,

%U 4181,6765,10946,17711,28657,46368,75025,121393,196418,317811,514229,832040

%N Fibonacci numbers (A000045) starting at offset -20.

%C 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

%H G. C. Greubel, <a href="/A147316/b147316.txt">Table of n, a(n) for n = -20..1000</a>

%H Philipp Fahr and Claus Michael Ringel, <a href="http://www.mathematik.uni-bielefeld.de/~ringel/opus/fr-zwei.pdf">Categorification of the Fibonacci Numbers Using Representations of Quivers</a>

%H <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (1,1).

%F a(n) = a(n-1) + a(n-2). - _R. J. Mathar_, Nov 30 2008

%F G.f.: (-6765 + 10946*x)/((1-x-x^2)*x^20). - _G. C. Greubel_, Jan 10 2020

%p with(combinat):seq(fibonacci(n), n=-20..30); # _G. C. Greubel_, Jan 10 2020

%t Array[Fibonacci, 51, -20] (* _Michael De Vlieger_, May 10 2017 *)

%t Fibonacci[Range[-20,30]] (* _G. C. Greubel_, Jan 10 2020 *)

%o (PARI) a(n)=fibonacci(n) \\ _M. F. Hasler_, May 10 2017

%o (Magma) [Fibonacci(n): n in [-20..30]]; // _G. C. Greubel_, Jan 10 2020

%o (Sage) [fibonacci(n) for n in (-20..50)] # _G. C. Greubel_, Jan 10 2020

%o (GAP) List([-20..30], n-> Fibonacci(n)); # _G. C. Greubel_, Jan 10 2020

%Y Cf. A000285, A022113, A001060.

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

%Y See also A039834 for A000045 extended to negative indices.

%K sign,easy,less

%O -20,1

%A _Roger L. Bagula_, Nov 05 2008

%E Extended to n = -20 .. 30 by _M. F. Hasler_, May 10 2017