login
This site is supported by donations to The OEIS Foundation.

 

Logo


Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
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. - From 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)

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 Rich Schroeppel, Somos Sequence Near-Addition Formulas and Modular Theta Functions, arXiv:math/0703470 [math.NT], 2007.

C. Kimberling, Strong divisibility sequences and some conjectures, Fib. Quart., 17 (1979), 13-17.

M. Somos, Number Walls in Combinatorics

Index entries for two-way infinite sequences

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[3] = -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-Fran├žois Alcover, Dec 14 2011, after first Pari program *)

PROG

(PARI) {a(n) = my(an); if( n<0, -a(-n), if( n==0, 0, an = vector( max(3, n), i, 1); an[3] = -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) / sqrt( -ellsigma(E, z) * ellsigma( E, 3 * z) / ellsigma( E, 2 * z)^2)^(n^2))}; /* Michael Somos, Oct 22 2004 */

(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

(PARI) {a(n) = subst( elldivpol( ellinit( [0, 0, -1, -1, 0]), n), x, 0)}; /* Michael Somos, Dec 16 2014 */

CROSSREFS

Cf. A006720, A028941, A050512, A051138, A178072, A278314.

Sequence in context: A217036 A127201 A225844 * A075643 A076074 A286390

Adjacent sequences:  A006766 A006767 A006768 * A006770 A006771 A006772

KEYWORD

sign,easy,nice

AUTHOR

Michael Somos, Jul 16 1999

STATUS

approved

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recent | More pages
The OEIS Community | Maintained by The OEIS Foundation Inc.

License Agreements, Terms of Use, Privacy Policy .

Last modified August 23 06:22 EDT 2017. Contains 290958 sequences.