A152163 a(n) = a(n-1)+a(n-2), n>1 ; a(0)=1, a(1)=-1.
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, -1346269, -2178309, -3524578, -5702887
Offset: 0
Links
- Vincenzo Librandi, Table of n, a(n) for n = 0..1000
- Index entries for linear recurrences with constant coefficients, signature (1,1).
Crossrefs
Cf. A000045.
Programs
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Magma
I:=[1, -1]; [n le 2 select I[n] else Self(n-1)+Self(n-2): n in [1..40]]; // Vincenzo Librandi, Feb 23 2013
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Mathematica
LinearRecurrence[{1,1},{1,-1},40] (* Harvey P. Dale, Oct 09 2012 *) -
PARI
a(n)=-fibonacci(n-2) \\ M. F. Hasler, May 10 2017
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Sage
def A152163(): a, b = True, False x, y = 1, 1 while True: yield x if a else -x x, y = y, x - y a, b = b, a a = A152163() print([next(a) for in range(50)]) # _Peter Luschny, Mar 19 2020
Formula
G.f.: (1-2*x)/(1-x-x^2).
a(n) = Sum_{k, 0<=k<=n}A147703(n,k)*(-2)^k.
a(n) = -Fibonacci(n-2) for n >= 2, and for all n if A000045 is extended in the natural way to negative indices; see also A039834. [Extended by M. F. Hasler, May 10 2017]
a(n) = (-1)^n*A039834(n-2). - R. J. Mathar, Mar 22 2011
G.f.: (1/(1-Q(0))-1)*(1-2*x)/x where Q(k)=1 - x^k/(1 - x/( x - x^k/Q(k+1) )); (continued fraction). - Sergei N. Gladkovskii, Feb 23 2013
G.f.: 2 - 2/(Q(0)+1) where Q(k) = 1 - 2*x/(1 - x/(x - 1/Q(k+1) )); (continued fraction ). - Sergei N. Gladkovskii, Apr 05 2013
G.f.: 1 - x - x^3*Q(0)/2, where Q(k) = 1 + 1/(1 - x*(6*k+1 + x)/(x*(6*k+4 + x) + 1/Q(k+1) )); (continued fraction). - Sergei N. Gladkovskii, Jan 02 2014
G.f.: 1+1/x - x - Q(0)/x, where Q(k) = 1 + x^2 - x^3 - k*x*(1+x^2) - x^2*( x*(k+2)-1)*( k*x -1)/Q(k+1) ; (continued fraction). - Sergei N. Gladkovskii, Jan 13 2014
E.g.f.: exp(x/2)*(5*cosh(sqrt(5)*x/2) - 3*sqrt(5)*sinh(sqrt(5)*x/2))/5. - Stefano Spezia, Mar 09 2025