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Search: id:A119282
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| A119282 |
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Alternating sum of the first n Fibonacci numbers. |
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+0
10
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| 0, -1, 0, -2, 1, -4, 4, -9, 12, -22, 33, -56, 88, -145, 232, -378, 609, -988, 1596, -2585, 4180, -6766, 10945, -17712, 28656, -46369, 75024, -121394, 196417, -317812, 514228, -832041, 1346268, -2178310, 3524577, -5702888, 9227464, -14930353, 24157816, -39088170, 63245985, -102334156, 165580140, -267914297, 433494436, -701408734, 1134903169, -1836311904, 2971215072, -4807526977, 7778742048
(list; graph; listen)
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OFFSET
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0,4
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COMMENT
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Apart from signs, same as A008346.
Natural bilateral extension (brackets mark index 0): ..., 88, 54, 33, 20, 12, 7, 4, 2, 1, 0, [0], -1, 0, -2, 1, -4, 4, -9, 12, -22, 3, ... This is A000071-reversed followed by A119282.
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FORMULA
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Let F(n) be the Fibonacci number A000045(n).
a(n) = sum_{k=1..n} (-1)^k F(k)
Closed form: a(n) = (-1)^n F(n-1) - 1 = (-1)^n A008346(n-1)
Recurrence: a(n) - 2 a(n-2) + a(n-3)= 0
G.f.: A(x) = -x/(1 - 2 x^2 + x^3) = -x/((1 - x)(1 + x - x^2))
Another recurrence: a(n) = a(n-2) - a(n-1) - 1. [From Rick L. Shepherd (rshepherd2(AT)hotmail.com), Aug 12 2009]
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MATHEMATICA
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a[n_Integer] := If[ n >= 0, Sum[ (-1)^k Fibonacci[k], {k, 1, n} ], Sum[ -(-1)^k Fibonacci[ -k], {k, 1, -n - 1} ] ]
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CROSSREFS
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Cf. A000071, A008346, A119283, A119284, A119285, A119286, A119287
Cf. A000071, A119283, A119284, A119285, A119286, A119287, A128696, A128698.
Sequence in context: A074763 A099932 A008346 this_sequence A095293 A034409 A048049
Adjacent sequences: A119279 A119280 A119281 this_sequence A119283 A119284 A119285
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KEYWORD
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sign,easy
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AUTHOR
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Stuart Clary (clary(AT)uakron.edu), May 13, 2006
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