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Search: id:A006769
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| A006769 |
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Elliptic divisibility sequence associated with y^2-y=x^3-x and the point (0,0).
(Formerly M0157)
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+0
7
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| 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; listen)
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OFFSET
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0,6
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COMMENT
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A Somos-4 sequence.
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REFERENCES
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G. Everest, A. van der Poorten, I. Shparlinski and T. Ward, Recurrence Sequences, Amer. Math. Soc., 2003; pp. 11 and 164.
C. Kimberling, Strong divisibility sequences and some conjectures, Fib. Quart., 17 (1979), 13-17.
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LINKS
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T. D. Noe, Table of n, a(n) for n=0..100
H. W. Braden, V. Z. Enolskii and A. N. W. Hone, Bilinear recurrences and addition formulae for hyperelliptic sigma functions
G. Everest, Elliptic Divisibility Sequences
M. Somos, Number Walls in Combinatorics
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FORMULA
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a(n)=(a(n-1)a(n-3)+a(n-2)^2)/a(n-4).
a(2n+1)=a(n+2)a(n)^3 - a(n-1)a(n+1)^3, a(2n)=a(n+2)a(n)a(n-1)^2 - a(n)a(n-2)a(n+1)^2.
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PROGRAM
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(PARI) a(n)=local(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)=local(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)=local(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 */
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CROSSREFS
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A006720(n)=(-1)^n*a(2n-3).
Squared terms give A028941.
Sequence in context: A137655 A058202 A127201 this_sequence A075643 A076074 A135017
Adjacent sequences: A006766 A006767 A006768 this_sequence A006770 A006771 A006772
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KEYWORD
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sign,easy,nice
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AUTHOR
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Michael Somos
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