|
| |
|
|
A119282
|
|
Alternating sum of the first n Fibonacci numbers.
|
|
12
| |
|
|
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; refs; listen; history; internal format)
|
|
|
|
OFFSET
| 0,4
|
|
|
COMMENTS
| 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.
Alternating sums of rows of the triangle in A141169. [Reinhard Zumkeller, Mar 22 2011]
|
|
|
LINKS
| Index to sequences with linear recurrences with constant coefficients, signature (0,2,-1)
Reinhard Zumkeller, Table of n, a(n) for n = 0..1000
|
|
|
FORMULA
| 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]
|
|
|
MATHEMATICA
| a[n_Integer] := If[ n >= 0, Sum[ (-1)^k Fibonacci[k], {k, 1, n} ], Sum[ -(-1)^k Fibonacci[ -k], {k, 1, -n - 1} ] ]
(* for the absolute values *) a[0] = 0; a[1] = 1; a[n_] := a[n] = a[n - 1] + a[n - 2] - (-1)^n; Array[a, 50, 0] [From Robert G. Wilson v (rgwv(AT)rgwv.com), Nov 07 2010]
|
|
|
CROSSREFS
| Cf. A000071, A008346, A119283 - A119287
Cf. A128696, A128698.
Sequence in context: A099932 A175000 A008346 * A095293 A034409 A048049
Adjacent sequences: A119279 A119280 A119281 * A119283 A119284 A119285
|
|
|
KEYWORD
| sign,easy
|
|
|
AUTHOR
| Stuart Clary (clary(AT)uakron.edu), May 13, 2006
|
| |
|
|
