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%I #173 Jul 18 2022 19:34:58
%S 1,1,0,1,-1,2,-3,5,-8,13,-21,34,-55,89,-144,233,-377,610,-987,1597,
%T -2584,4181,-6765,10946,-17711,28657,-46368,75025,-121393,196418,
%U -317811,514229,-832040,1346269,-2178309,3524578,-5702887,9227465,-14930352,24157817
%N a(n+2) = -a(n+1) + a(n) (signed Fibonacci numbers) with a(-2) = a(-1) = 1; or Fibonacci numbers (A000045) extended to negative indices.
%C Knuth defines the negaFibonacci numbers as follows: F(-1) = 1, F(-2) = -1, F(-3) = 2, F(-4) = -3, F(-5) = 5, ..., F(-n) = (-1)^(n-1) F(n). See A215022, A215023 for the negaFibonacci representation of n. - _N. J. A. Sloane_, Aug 03 2012
%C The ratio of successive terms converges to -1/phi. - _Jonathan Vos Post_, Dec 10 2006
%C The sequence a(n), n >= 0 := 0, 1, -1, 2, -3, 5, -8, 13, ... is the inverse binomial transform of A000045. - _Philippe Deléham_, Oct 28 2008
%C Equals the INVERTi transform of A038754, assuming that an additional A038754(0) = 1 is added in front of A038754, and that the a(n) are prefixed with another 1 and then get offset 0. - _Gary W. Adamson_, Jan 08 2011
%C If we remove a(-2) and then set the offset to 0, we have the INVERT transform of a signed A011782: (1, -1, 2, -4, 8, -16, 32, ...).- _Gary W. Adamson_, Jan 08 2011
%C The sequence 0, 1, -1, 2, -3, 5, -8, 13, -21, 34, -55, 89, -144, ... (starting at offset 0) is the Lucas U(-1,-1) sequence. - _R. J. Mathar_, Jan 08 2013
%C This sequence appears in the formula for 1/rho(5)^n, with rho(5) = (1 + sqrt(5))/2 = phi (golden section), when written in the power basis <1, rho(5)> of the quadratic number field Q(rho(5)): 1/rho(5)^n = a(n+1) * 1 + a(n) * rho(5), n >= -2. - _Wolfdieter Lang_, Nov 04 2013
%C a(n) = A227431(n + 4, n + 3). - _Reinhard Zumkeller_, Feb 01 2014
%C The sequence 1, -1, 2, -3, 5, -8, 13, -21, 34, -55, 89, -144, ... (starting at offset 1) is the reversion of the g.f. for the "shadows" of Motzkin numbers with offset 1 (see A343773). - _Gennady Eremin_, Jul 16 2021
%D D. E. Knuth, The Art of Computer Programming, Vol. 4A, Section 7.1.3, p. 168, Eq. (145).
%D D. Shtefan and I. Dobrovolska, The sums of the consecutive Fibonacci numbers, Fib. Q., 56 (2018), 229-236.
%H Indranil Ghosh, <a href="/A039834/b039834.txt">Table of n, a(n) for n = -2..4773</a> (terms -2..500 from T. D. Noe)
%H Gennady Eremin, <a href="https://arxiv.org/abs/2108.10676">Walking in the OEIS: From Motzkin numbers to Fibonacci numbers. The "shadows" of Motzkin numbers</a>, arXiv:2108.10676 [math.CO], 2021.
%H M. Cetin Firengiz and A. Dil, <a href="http://www.nntdm.net/papers/nntdm-20/NNTDM-20-4-21-32.pdf">Generalized Euler-Seidel method for second order recurrence relations</a>, Notes on Number Theory and Discrete Mathematics, Vol. 20, 2014, No. 4, 21-32.
%H Jiřı Jina and Pavel Trojovský, <a href="http://dx.doi.org/10.12732/ijpam.v88i4.11">On determinants of some tridiagonal matrices connected with Fibonacci numbers</a>, International Journal of Pure and Applied Mathematics, Volume 88 No. 4 2013, 569-575; ISSN: 1311-8080 (printed version); ISSN: 1314-3395 (on-line version).
%H J. Pan, <a href="https://cs.uwaterloo.ca/journals/JIS/OL13/Pan/pan8.html">Multiple Binomial Transforms and Families of Integer Sequences </a>, J. Int. Seq. 13 (2010), 10.4.2.
%H J. Pan, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL14/Pan/pan12.html"> Some Properties of the Multiple Binomial Transform and the Hankel Transform of Shifted Sequences </a>, J. Int. Seq. 14 (2011) # 11.3.4, remark 14.
%H Emil Daniel Schwab and Gabriela Schwab, <a href="http://math.colgate.edu/~integers/wg3/wg3.pdf">k-Fibonacci numbers and Möbius Functions</a>, Integers (2022) Vol. 22, #A64.
%H Kai Wang, <a href="https://www.researchgate.net/publication/337943524_Fibonacci_Numbers_And_Trigonometric_Functions_Outline">Fibonacci Numbers And Trigonometric Functions Outline</a>, (2019).
%H Wikipedia, <a href="http://en.wikipedia.org/wiki/Lucas_sequence#Specific_names">Lucas sequence</a>
%H <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (-1,1).
%H <a href="/index/Lu#Lucas">Index entries for Lucas sequences</a>
%F G.f.: (1+2*x)/(x^2*(1+x-x^2)).
%F a(n-2) = Sum_{k=0..n} (-2)^k*A055830(n, k). - _Philippe Deléham_, Oct 18 2006
%F a(n) = ((phi - 1)^n + 1/phi*(-(1/phi) - 1)^(n+1))/sqrt(5), where phi = (1 + sqrt(5))/2. - _Arkadiusz Wesolowski_, Oct 28 2012
%F a(n) = Sum_{k = 1..n} binomial(n-1, k-1)*Fibonacci(k)*(-1)^(n-k), n > 0, a(0) = 1. - _Perminova Maria_, Jan 22 2013
%F G.f.: 1 + x/(Q(0) - x) where Q(k) = 1 - x/(x*k - 1)/Q(k+1); (continued fraction). - _Sergei N. Gladkovskii_, Feb 23 2013
%F 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
%F G.f.: 1 + x^2 + x^3 + x/Q(0), where Q(k) = 1 + (k+1)*x/(1 - x/(x + (k+1)/Q(k+1))); (continued fraction). - _Sergei N. Gladkovskii_, Apr 23 2013
%F G.f.: 1/(G(0)*x^3) + (2*x^2+x-1)/x^3, where G(k) = 1 + 2*x*(k+1)/(k + 2 - x*(k+2)*(k+3)/(x*(k+3) + (k+1)/G(k+1))); (continued fraction). - _Sergei N. Gladkovskii_, Jun 27 2013
%F G.f.: Q(0)/x - 1/x + 1+ x, where Q(k) = 1 + x^2 + x^3 + k*x*(1+x^2) - x^2*(1 + x*(k+2))*(1+k*x)/Q(k+1); (continued fraction). - _Sergei N. Gladkovskii_, Jan 13 2014
%F a(n) = -(-1)^n*A000045(n), at least for all n >= 0 (and also for n < 0 if A000045 is extended to negative indices). - _M. F. Hasler_, May 10 2017
%e From _Wolfdieter Lang_, Nov 04 2013: (Start)
%e With the golden section phi = rho(5) = (1 + sqrt(5))/2:
%e n = -2: phi^2 = a(-1)*1 + a(-2)*phi = 1 + phi,
%e n = -1: phi = a(0)*1 + a(-1)*phi = phi, (trivial)
%e n = 0: 1/phi^0 = a(1)*1 + a(0)*phi = 1, (trivial)
%e n = 1: 1/phi = a(2)*1 + a(1)*phi = -1 + phi,
%e n = 2: 1/phi^2 = a(3)*1 + a(2)*phi = 2 - phi, ... (End)
%e G.f. = x^-2 + x^-1 + x - x^2 + 2*x^3 - 3*x^4 + 5*x^5 - 8*x^6 + 13*x^7 - ...
%p a:= n-> (Matrix([[0, 1], [1, -1]])^n) [1,2]: seq(a(n), n=-2..50); # _Alois P. Heinz_, Nov 01 2008
%t LinearRecurrence[{-1, 1}, {1, 1}, 60] (* _Vladimir Joseph Stephan Orlovsky_, May 25 2011 *)
%t Fibonacci[-Range[-2, 37]] (* _Michael Somos_, Jun 04 2016 *)
%o (PARI) a(n) = fibonacci(-n);
%o (Haskell)
%o a039834 n = a039834_list !! (n+2)
%o a039834_list = 1 : 1 : zipWith (-) a039834_list (tail a039834_list)
%o -- _Reinhard Zumkeller_, Jul 05 2013
%o (Sage)
%o def A039834():
%o x, y = 1, 1
%o while True:
%o yield x
%o x, y = y, x - y
%o a = A039834()
%o [next(a) for i in range(40)] # _Peter Luschny_, Jul 11 2013
%o (Sage)
%o def A039834_list(len):
%o R.<t> = LaurentSeriesRing(ZZ, 't', default_prec = len)
%o f = (-2*t-1)/(t^4-t^3-t^2)
%o return f.list()
%o A039834_list(40) # _Peter Luschny_, Nov 21 2014
%o (Magma) [Fibonacci(-n): n in [-2..40]]; // _Marius A. Burtea_, Nov 14 2019
%o (Python)
%o from sympy import fibonacci
%o def A039834(n): return fibonacci(-n) # _Chai Wah Wu_, Jan 20 2022
%Y Cf. A000045, A001622, A038754, A011782, A055830, A215022, A215023, A343773.
%K sign,easy,nice
%O -2,6
%A Alexander Grasser (pyropunk(AT)usa.net)
%E Signs corrected by _Len Smiley_ and _N. J. A. Sloane_
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