A039834 - OEIS login The OEIS is supported by the many generous donors to the OEIS Foundation. Hints (Greetings from The On-Line Encyclopedia of Integer Sequences!) A039834 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. 62 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, 9227465, -14930352, 24157817, -39088169 (list; graph; refs; listen; history; text; internal format) OFFSET -2,6 COMMENTS 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 The ratio of successive terms converges to -1/phi. - Jonathan Vos Post, Dec 10 2006 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 From Gary W. Adamson, Jan 08 2011: (Start) 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. 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, ...). (End) 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 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 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 A Pisano sequence (modulo m) terminates with (..., 13, -8, 5, -3, 2, -1, 1, 0). - Roman Stawski, Nov 11 2024 REFERENCES Donald E. Knuth, The Art of Computer Programming, Vol. 4A, Section 7.1.3, p. 168, Eq. (145). LINKS Indranil Ghosh, Table of n, a(n) for n = -2..4773 (terms -2..500 from T. D. Noe) Gaurav Bhatnagar, Analogues of a Fibonacci-Lucas Identity, Fib. Quart. 54(2) (2016), 166-171. Gennady Eremin, Walking in the OEIS: From Motzkin numbers to Fibonacci numbers. The "shadows" of Motzkin numbers, arXiv:2108.10676 [math.CO], 2021. M. Cetin Firengiz and A. Dil, Generalized Euler-Seidel method for second order recurrence relations, Notes Num. Theory Disc. Math. 20(4) (2014), 21-32. Jiřı Jina and Pavel Trojovský, On determinants of some tridiagonal matrices connected with Fibonacci numbers, Int'l J. Pure Appl. Math. 88(4) (2013), 569-575; ISSN: 1311-8080 (printed version); ISSN: 1314-3395 (on-line version). Peter Luschny, The oscillatory Binet continuation of the Fibonacci numbers. Ivica Martinjak, Two Extensions of the Sury's Identity, arXiv:1508.01444 [math.CO], 2015. Jiaqiang Pan, Multiple Binomial Transforms and Families of Integer Sequences, J. Int. Seq. 13 (2010), Art. 10.4.2. Jiaqiang Pan, Some Properties of the Multiple Binomial Transform and the Hankel Transform of Shifted Sequences, J. Int. Seq. 14 (2011), Art. 11.3.4. See Remark 14. Emil Daniel Schwab and Gabriela Schwab, k-Fibonacci numbers and Möbius Functions, Integers 22 (2022), Art. A64. Dmitriy Shtefan and Irina Dobrovolska, The sums of the consecutive Fibonacci numbers, Fib. Quart. 56 (2018), 229-236. Kai Wang, Fibonacci Numbers And Trigonometric Functions Outline, (2019). Wikipedia, Lucas sequence. Fan Zhu, Song He, and Zhenqi Han, Planar loop integrands from cuts in D dimensions, arXiv:2606.28052 [hep-th], 2026. See p. 22 (Fig. 13). Index entries for linear recurrences with constant coefficients, signature (-1,1). Index entries for Lucas sequences. FORMULA G.f.: (1+2*x)/(x^2*(1+x-x^2)). a(n-2) = Sum_{k=0..n} (-2)^k*A055830(n, k). - Philippe Deléham, Oct 18 2006 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 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 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 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^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 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 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 a(n) = A227431(n + 4, n + 3). - Reinhard Zumkeller, Feb 01 2014 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 a(n) = Sum_{k=0..floor((n-1)/2)} A130595(n-k-1, k), for n >= 0. - G. C. Greubel, Jun 22 2024 From Amiram Eldar, Jan 10 2026: (Start) a(n) = Sum_{k=0..n-1} (-1)^k * 2^(n-1-k) * Lucas(k+1) for n >= 0, where Lucas(k) = A000032(k) (Martinjak, 2015). a(n) = Sum_{k=0..n-1} (-1)^k * (Lucas(k+1) - Fibonacci(k)) for n >= 0 (Bhatnagar , 2016). (End) From Peter Luschny, May 24 2026: (Start) a(n) = (phi^(-n) - phi^(n)*exp(-I*Pi*n))/sqrt(5) where phi = (1 + sqrt(5))/2. Written in this form the function a(x) satisfies the Cassini analog: a(x + 1) * a(x - 1) - a(x)^2 = exp(-I*Pi*x). (End) EXAMPLE From Wolfdieter Lang, Nov 04 2013: (Start) With the golden section phi = rho(5) = (1 + sqrt(5))/2: n = -2: phi^2 = a(-1)*1 + a(-2)*phi = 1 + phi, n = -1: phi = a(0)*1 + a(-1)*phi = phi, (trivial) n = 0: 1/phi^0 = a(1)*1 + a(0)*phi = 1, (trivial) n = 1: 1/phi = a(2)*1 + a(1)*phi = -1 + phi, n = 2: 1/phi^2 = a(3)*1 + a(2)*phi = 2 - phi, ... (End) G.f. = x^-2 + x^-1 + x - x^2 + 2*x^3 - 3*x^4 + 5*x^5 - 8*x^6 + 13*x^7 - ... MAPLE a:= n-> (Matrix([[0, 1], [1, -1]])^n) [1, 2]: seq(a(n), n=-2..50); # Alois P. Heinz, Nov 01 2008 MATHEMATICA LinearRecurrence[{-1, 1}, {1, 1}, 60] (* Vladimir Joseph Stephan Orlovsky, May 25 2011 *) Fibonacci[-Range[-2, 37]] (* Michael Somos, Jun 04 2016 *) PROG (PARI) a(n) = fibonacci(-n); (Haskell) a039834 n = a039834_list !! (n+2) a039834_list = 1 : 1 : zipWith (-) a039834_list (tail a039834_list) -- Reinhard Zumkeller, Jul 05 2013 (SageMath) def A039834(): x, y = 1, 1 while True: yield x x, y = y, x - y a = A039834() [next(a) for i in range(40)] # Peter Luschny, Jul 11 2013 (SageMath) def A039834_list(len): R.<t> = LaurentSeriesRing(ZZ, 't', default_prec = len) f = (-2*t-1)/(t^4-t^3-t^2) return f.list() A039834_list(40) # Peter Luschny, Nov 21 2014 (Magma) [Fibonacci(-n): n in [-2..40]]; // Marius A. Burtea, Nov 14 2019 (Python) from sympy import fibonacci def A039834(n): return fibonacci(-n) # Chai Wah Wu, Jan 20 2022 CROSSREFS Cf. A000032, A000045, A001622, A038754, A011782, A055830, A130595, A215022, A215023, A227431, A343773. Sequence in context: A185357 A132636 A152163 * A236191 A333378 A000045 Adjacent sequences: A039831 A039832 A039833 * A039835 A039836 A039837 KEYWORD sign,easy,nice AUTHOR Alexander Grasser (pyropunk(AT)usa.net) EXTENSIONS Signs corrected by Len Smiley and N. J. A. 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