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 A006769 Elliptic divisibility sequence associated with elliptic curve "37a1": y^2 + y = x^3 - x and multiples of the point (0,0). (Formerly M0157) 11
 0, 1, 1, -1, 1, 2, -1, -3, -5, 7, -4, -23, 29, 59, 129, -314, -65, 1529, -3689, -8209, -16264, 83313, 113689, -620297, 2382785, 7869898, 7001471, -126742987, -398035821, 1687054711, -7911171596, -47301104551, 43244638645 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,6 COMMENTS This sequence has a recursion same as the Somos-4 sequence recursion. a(n+1) is the Hankel transform of A178072. - Paul Barry, May 19 2010 The recurrence formulas in [Kimberling, p. 16] are missing square and cube exponents. - Michael Somos, Jul 07 2014 This is a strong elliptic divisibility sequence t_n as given in [Kimberling, p. 16] where x = 1, y = -1, z = 1. REFERENCES G. Everest, A. van der Poorten, I. Shparlinski and T. Ward, Recurrence Sequences, Amer. Math. Soc., 2003; pp. 11 and 164. N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence). LINKS T. D. Noe and Seiichi Manyama, Table of n, a(n) for n = 0..300 (first 101 terms from T. D. Noe) Paul Barry, Riordan Pseudo-Involutions, Continued Fractions and Somos 4 Sequences, arXiv:1807.05794 [math.CO], 2018. Paul Barry, Generalized Catalan recurrences, Riordan arrays, elliptic curves, and orthogonal polynomials, arXiv:1910.00875 [math.CO], 2019. Paul Barry, Riordan arrays, the A-matrix, and Somos 4 sequences, arXiv:1912.01126 [math.CO], 2019. H. W. Braden, V. Z. Enolskii and A. N. W. Hone, Bilinear recurrences and addition formulas for hyperelliptic sigma functions, arXiv:math/0501162 [math.NT], 2005. Graham Everest, Elliptic Divisibility Sequences. R. W. Gosper and Richard C. Schroeppel, Somos Sequence Near-Addition Formulas and Modular Theta Functions, arXiv:math/0703470 [math.NT], 2007. Clark Kimberling, Strong divisibility sequences and some conjectures, Fib. Quart., 17 (1979), 13-17. Michael Somos, Number Walls in Combinatorics FORMULA a(n) = (a(n-1) * a(n-3) + a(n-2)^2) / a(n-4) for all n in Z except n=4. a(n) = (-a(n-1) * a(n-4) - a(n-2) * a(n-3)) / a(n-5) for all n in Z except n=5. a(-n) = -a(n) for all n in Z. a(2*n + 1) = a(n+2) * a(n)^3 - a(n-1) * a(n+1)^3, a(2*n) = a(n+2) * a(n) * a(n-1)^2 - a(n) * a(n-2) * a(n+1)^2 for all n in Z. A006720(n) = (-1)^n * a(2*n - 3), A028941(n) = a(n)^2 for all n in Z. a(2*n) = A051138(n). - Michael Somos, Feb 10 2015 MATHEMATICA a[n_] := If[n < 0, -a[-n], If[n == 0, 0, ClearAll[an]; an[_] = 1; an = -1; For[k = 5, k <= n, k++, an[k] = (an[k-1]*an[k-3] + an[k-2]^2)/an[k-4]]; an[n]]]; Table[a[n], {n, 0, 32}] (* Jean-François Alcover, Dec 14 2011, after first Pari program *) Join[{0}, RecurrenceTable[{a==a==1, a==-1, a==1, a[n]==(a[n-1] a[n-3]+ a[n-2]^2)/a[n-4]}, a, {n, 40}]] (* Harvey P. Dale, May 04 2018 *) PROG (PARI) {a(n) = my(an); if( n<0, -a(-n), if( n==0, 0, an = vector( max(3, n), i, 1); an = -1; for( k=5, n, an[k] = (an[k-1] * an[k-3] + an[k-2]^2) / an[k-4]); an[n]))}; (PARI) {a(n) = my(an); if( n<0, -a(-n), if( n==0, 0, an = Vec((-1 - 2*x + sqrt(1 + 4*x - 4*x^3 + O(x^n))) / (2 * x^2)); matdet( matrix((n-1)\2, (n-1)\2, i, j, if(i + j - 1 - n%2<0, 0, an[i + j -n%2])))))}; (PARI) {a(n) = my(E, z); E = ellinit([0, 0, -1, -1, 0]); z = ellpointtoz(E, [0, 0]); round( ellsigma(E, n*z) / ellsigma(E, z)^(n^2))}; /* Michael Somos, Oct 22 2004 */ (PARI) {a(n) = sign(n) * subst( elldivpol( ellinit([0, 0, -1, -1, 0]), abs(n)), x, 0)}; /* Michael Somos, Dec 16 2014 */ (Haskell) a006769 n = a050512_list !! n a006769_list = 0 : 1 : 1 : (-1) : 1 : zipWith div (zipWith (+) (zipWith (*)    (drop 4 a006769_list) (drop 2 a006769_list))      (map (^ 2) (drop 3 a006769_list))) (tail a006769_list) -- Reinhard Zumkeller, Nov 02 2011 CROSSREFS Cf. A006720, A028941, A050512, A051138, A178072, A278314. Sequence in context: A217036 A127201 A225844 * A075643 A076074 A319153 Adjacent sequences:  A006766 A006767 A006768 * A006770 A006771 A006772 KEYWORD sign,easy,nice AUTHOR Michael Somos, Jul 16 1999 STATUS approved

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Last modified December 4 18:29 EST 2020. Contains 338936 sequences. (Running on oeis4.)