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### -- Authors: Tobias Boege
### -- File: t/data/Fibonacci-Search-all.txt
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### -- Chunk 15 of 15
 for the negaFibonacci representation of n. - _N. J. A. Sloane_, Aug 03 2012
%C A039834 The ratio of successive terms converges to -1/phi. - _Jonathan Vos Post_, Dec 10 2006
%C A039834 Let a(n) := F(n) * (-1)^binom(n, 2). Then a(m - n) * a(m + n) = a(m + 1) * a(m - 1) * a(n)^2 - a(n + 1) * a(n - 1) * a(m)^2. This plus gcd(f[n], f[m]) = |f[gcd(n, m)]| makes a[] a strong elliptic divisibility sequence. Likewise F(n) * (-1)^binom(n - 1, 2), but no other asSIGNation (mod scaling). - _Bill Gosper_, May 28 2008
%C A039834 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 A039834 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 A039834 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 A039834 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 A039834 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 A039834 a(n) = A227431(n + 4, n + 3). - _Reinhard Zumkeller_, Feb 01 2014
%D A039834 D. E. Knuth, The Art of Computer Programming, Vol. 4A, Section 7.1.3, p. 168, Eq. (145).
%D A039834 D. Shtefan and I. Dobrovolska, The sums of the consecutive Fibonacci numbers, Fib. Q., 56 (2018), 229-236.
%H A039834 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 A039834 M. Cetin Firengiz, 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 A039834 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 A039834 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 A039834 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 A039834 Wikipedia, <a href="http://en.wikipedia.org/wiki/Lucas_sequence#Specific_names">Lucas sequence</a>
%H A039834 <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (-1,1)
%H A039834 <a href="/index/Lu#Lucas">Index entries for Lucas sequences</a>
%F A039834 G.f.: (1+2*x)/(x^2*(1+x-x^2)).
%F A039834 a(n-2) = Sum_{k, 0 <= k <= n}(-2)^k*A055830(n, k). - _Philippe Deléham_, Oct 18 2006
%F A039834 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 A039834 a(n) = sum(k = 1..n, binomial(n - 1, k - 1)*fib(k)*(-1)^(n - k)), n > 0, fib(k) = A000045(k), a(0) = 1. - _Perminova Maria_, Jan 22 2013
%F A039834 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 A039834 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 A039834 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 A039834 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 A039834 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 A039834 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 A039834 From - _Wolfdieter Lang_, Nov 04 2013 (Start)
%e A039834 With the golden section phi = rho(5) = (1 + sqrt(5))/2:
%e A039834 n = -2: phi^2 = a(-1)*1 + a(-2)*phi = 1 + phi,
%e A039834 n = -1: phi = a(0)*1 + a(-1)*phi = phi, (trivial)
%e A039834 n =  0: 1/phi^0 =  a(1)*1 + a(0)*phi = 1, (trivial)
%e A039834 n =  1: 1/phi = a(2)*1 + a(1)*phi = -1 + phi,
%e A039834 n =  2: 1/phi^2 = a(3)*1 + a(2)*phi = 2 - phi, ... (End)
%e A039834 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 A039834 a:= n-> (Matrix([[0, 1], [1, -1]])^n) [1,2]: seq(a(n), n=-2..50); # _Alois P. Heinz_, Nov 01 2008
%t A039834 LinearRecurrence[{-1, 1}, {1, 1}, 60] (* _Vladimir Joseph Stephan Orlovsky_, May 25 2011 *)
%t A039834 Fibonacci[-Range[-2, 37]] (* _Michael Somos_, Jun 04 2016 *)
%o A039834 (PARI) a(n) = fibonacci(-n);
%o A039834 (Haskell)
%o A039834 a039834 n = a039834_list !! (n+2)
%o A039834 a039834_list = 1 : 1 : zipWith (-) a039834_list (tail a039834_list)
%o A039834 -- _Reinhard Zumkeller_, Jul 05 2013
%o A039834 (Sage)
%o A039834 def A039834():
%o A039834     x, y = 1, 1
%o A039834     while true:
%o A039834         yield x
%o A039834         x, y = y, x - y
%o A039834 a = A039834()
%o A039834 [a.next() for i in range(40)]  # _Peter Luschny_, Jul 11 2013
%o A039834 (Sage)
%o A039834 def A039834_list(len):
%o A039834     R.<t> = LaurentSeriesRing(ZZ, 't', default_prec = len)
%o A039834     f = (-2*t-1)/(t^4-t^3-t^2)
%o A039834     return f.list()
%o A039834 A039834_list(40) # _Peter Luschny_, Nov 21 2014
%Y A039834 Cf. A000045, A038754, A011782, A215022, A215023.
%K A039834 sign,easy,nice
%O A039834 -2,6
%A A039834 Alexander Grasser (pyropunk(AT)usa.net)
%E A039834 Signs corrected by _Len Smiley_ and _N. J. A. Sloane_

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# Greetings from The On-Line Encyclopedia of Integer Sequences! http://oeis.org/

Search: seq:1,1,2,3,5,8,13,21
Showing 91-92 of 92

%I A152163
%S A152163 1,-1,0,-1,-1,-2,-3,-5,-8,-13,-21,-34,-55,-89,-144,-233,-377,-610,
%T A152163 -987,-1597,-2584,-4181,-6765,-10946,-17711,-28657,-46368,-75025,
%U A152163 -121393,-196418,-317811,-514229,-832040,-1346269,-2178309,-3524578,-5702887
%N A152163 a(n)=a(n-1)+a(n-2), n>1 ; a(0)=1, a(1)=-1 .
%H A152163 Vincenzo Librandi, <a href="/A152163/b152163.txt">Table of n, a(n) for n = 0..1000</a>
%H A152163 <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (1,1).
%F A152163 G.f.: (1-2*x)/(1-x-x^2).
%F A152163 a(n) = Sum_{k, 0<=k<=n}A147703(n,k)*(-2)^k.
%F A152163 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]
%F A152163 a(n) = (1/2)*{[(1/2)+(1/2)*sqrt(5)]^n+[(1/2)-(1/2)*sqrt(5)]^n}+(3/10)*sqrt(5)*{[(1/2)-(1/2) *sqrt(5)]^n-[(1/2)+(1/2)*sqrt(5)]^n}, with n>=0 [From _Paolo P. Lava_, Dec 01 2008]
%F A152163 a(n) = (-1)^n*A039834(n-2). - _R. J. Mathar_, Mar 22 2011
%F A152163 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
%F A152163 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 A152163 a(n) = A000045(n+1)-2*A000045(n). - _R. J. Mathar_, Jun 26 2013
%F A152163 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
%F A152163 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
%t A152163 LinearRecurrence[{1,1},{1,-1},40] (* _Harvey P. Dale_, Oct 09 2012 *)
%o A152163 (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
%o A152163 (PARI) a(n)=-fibonacci(n-2) \\ _M. F. Hasler_, May 10 2017
%Y A152163 Cf. A000045.
%K A152163 easy,sign
%O A152163 0,6
%A A152163 _Philippe Deléham_, Nov 27 2008

%I A236191
%S A236191 0,1,1,2,-3,-5,-8,13,21,34,-55,-89,-144,233,377,610,-987,-1597,-2584,
%T A236191 4181,6765,10946,-17711,-28657,-46368,75025,121393,196418,-317811,
%U A236191 -514229,-832040,1346269,2178309,3524578,-5702887,-9227465,-14930352,24157817,39088169
%N A236191 (-1)^floor( (n-1) / 3 ) * F(n), where F = Fibonacci.
%H A236191 Vincenzo Librandi, <a href="/A236191/b236191.txt">Table of n, a(n) for n = 0..1000</a>
%H A236191 <a href="/index/Rec">Index entries for linear recurrences with constant coefficients</a>, signature (0,0,-4,0,0,1).
%F A236191 G.f.: (x + x^2 + 2*x^3 + x^4 - x^5) / (1 + 4*x^3 - x^6).
%F A236191 a(n) = -4 * a(n-3) + a(n-6). a(-n) = (-1)^floor( (n-2) / 3) * a(n) for all n in Z.
%F A236191 a(n) * a(n+3) = a(n+1)^2 - a(n+2)^2 for all n in Z.
%e A236191 G.f. = x + x^2 + 2*x^3 - 3*x^4 - 5*x^5 - 8*x^6 + 13*x^7 + 21*x^8 + ...
%t A236191 a[ n_] := (-1)^Quotient[n-1, 3] Fibonacci[n];
%t A236191 CoefficientList[Series[(x + x^2 + 2 x^3 + x^4 - x^5)/(1 + 4 x^3 - x^6), {x, 0, 50}], x] (* _Vincenzo Librandi_, Jan 20 2014 *)
%o A236191 (PARI) {a(n) = (-1)^( (n-1) \ 3) * fibonacci( n)};
%o A236191 (MAGMA) I:=[0,1,1,2,-3,-5]; [n le 6 select I[n] else -4*Self(n-3)+Self(n-6): n in [1..40]]; // _Vincenzo Librandi_, Jan 20 2014
%Y A236191 Cf. A000045.
%K A236191 sign,easy
%O A236191 0,4
%A A236191 _Michael Somos_, Jan 19 2014

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