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A147316
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Fibonacci numbers (A000045) starting at offset -20.
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4
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-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
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-20,1
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COMMENTS
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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
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LINKS
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FORMULA
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G.f.: (-6765 + 10946*x)/((1-x-x^2)*x^20). - G. C. Greubel, Jan 10 2020
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MAPLE
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with(combinat):seq(fibonacci(n), n=-20..30); # G. C. Greubel, Jan 10 2020
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MATHEMATICA
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PROG
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(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
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CROSSREFS
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Of course A000045 is the main entry for the Fibonacci numbers.
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KEYWORD
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sign,easy,less
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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