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A006720
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Somos-4 sequence: a(0)=a(1)=a(2)=a(3)=1; for n >= 4, a(n)=(a(n-1)a(n-3)+a(n-2)^2)/a(n-4).
(Formerly M0857)
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61
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1, 1, 1, 1, 2, 3, 7, 23, 59, 314, 1529, 8209, 83313, 620297, 7869898, 126742987, 1687054711, 47301104551, 1123424582771, 32606721084786, 1662315215971057, 61958046554226593, 4257998884448335457, 334806306946199122193
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OFFSET
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0,5
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COMMENTS
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From the 5th term on, all terms have a primitive divisor; in other words, a prime divisor that divides no earlier term in the sequence. A proof appears in the Everest-McLaren-Ward paper. - Graham Everest (g.everest(AT)uea.ac.uk), Oct 26 2005
Twelve prime terms are known, occurring at indices 4, 5, 6, 7, 8, 11, 13, 16, 43, 52, 206, 647. The last two have been checked for probable primality only. The 647-th term has 18498 decimal digits. Possibly these are the only prime terms in the entire sequence. - Graham Everest (g.everest(AT)uea.ac.uk), Nov 28 2006
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REFERENCES
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P. Barry, Riordan-Bernstein Polynomials, Hankel Transforms and Somos Sequences, Journal of Integer Sequences, Vol. 15 2012, #12.8.2. - From N. J. A. Sloane, Dec 29 2012
R. H. Buchholz and R. L. Rathbun, "An infinite set of Heron triangles with two rational medians", Amer. Math. Monthly, 104 (1997), 107-115.
G. Everest, A. van der Poorten, I. Shparlinski and T. Ward, Recurrence Sequences, Amer. Math. Soc., 2003; pp. 9, 179.
G. Everest et al., Primes generated by recurrence sequences, Amer. Math. Monthly, 114 (No. 5, 2007), 417-431.
David Gale, Mathematical Entertainments: "The strange and surprising saga of the Somos sequences", Math. Intelligencer, 13(1) (1991), pp. 40-42.
A. N. W. Hone, Elliptic curves and quadratic recurrence sequences, Bulletin of the London Mathematical Society 37 (2005) 161-171.
J. L. Malouf, "An integer sequence from a rational recursion", Discr. Math. 110 (1992), 257-261.
R. M. Robinson, "Periodicity of Somos sequences", Proc. Amer. Math. Soc., 116 (1992), 613-619.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
Alfred J. van der Poorten, Elliptic Curves and Continued Fractions, Journal of Integer Sequences, Vol. 8 (2005), Article 05.2.5.
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LINKS
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Robert G. Wilson v, Table of 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
S. B. Ekhad and D. Zeilberger, How To Generate As Many Somos-Like Miracles as You Wish, arXiv preprint arXiv:1303.5306, 2013
Graham Everest, Gerard Mclaren and Tom Ward, Primitive divisors of elliptic divisibility sequences 2005
G. Everest, S. Stevens, D. Tamsett and T. Ward, Primitive divisors of quadratic polynomial sequences
S. Fomin and A. Zelevinsky, The Laurent phenomemon
Allan Fordy and Andrew Hone, Discrete integrable systems and Poisson algebras from cluster maps, Arxiv preprint arXiv:1207.6072, 2012.
A. N. W. Hone, Sigma function solution of the initial value problem for Somos 5 sequences.
A. N. W. Hone, Algebraic curves, integer sequences and a discrete Painleve transcendent, Proceedings of SIDE 6, Helsinki, Finland, 2004.
J. Propp, The Somos Sequence Site
J. Propp, The 2002 REACH tee-shirt
Vladimir Shevelev and Peter J. C. Moses, On a sequence of polynomials with hypothetically integer coefficients, arXiv preprint arXiv:1112.5715, 2011
M. Somos, Somos 6 Sequence
M. Somos, Brief history of the Somos sequence problem
D. E. Speyer, Perfect matchings and the octahedral recurrence
A. J. van der Poorten, Recurrence relations for elliptic sequences...
A. J. van der Poorten, Hyperelliptic curves, continued fractions and Somos sequences
Eric Weisstein's World of Mathematics, Somos Sequence
Index entries for two-way infinite sequences
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FORMULA
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a(n+1)/a(n) seems to be asymptotic to C^n with C=1.226....... - Benoit Cloitre, Aug 07 2002. Confirmed by Hone - see below.
The terms of the sequence have the leading order asymptotics log a(n) ~ D n^2 with D = zeta(w1)*k^2/(2*w1)-log|sigma(k)| = 0.10222281... where zeta and sigma are the Weierstrass functions with invariants g2 = 4, g3 = -1, w1 = 1.496729323 is the real half-period of the corresponding elliptic curve, k = -1.134273216 as above. This agrees with Benoit Cloitre's numerical result with C = exp(2D) = 1.2268447... - Andrew Hone (anwh(AT)kent.ac.uk), Feb 09 2005
a(n) = (a(n-1)*a(n-3) + a(n-2)^2)/a(n-4); a(0) = a(1) = a(2) = a(3) = 1; exact formula is a(n) = A*B^n*sigma (z_0+nk)/(sigma (k))^(n^2), where sigma is the Weierstrass sigma function associated to the elliptic curve y^2 = 4*x^3-4*x+1, A = 1/sigma(z_0) = 0.112724016-0.824911687*i, B = sigma(k)*sigma (z_0)/sigma (z_0+k) = 0.215971963+0.616028193*i, k = 1.859185431, z_0 = 0.204680500+1.225694691*i, sigma(k) = 1.555836426, all to 9 decimal places. This is a special case of a general formula for 4th order bilinear recurrences. The Somos-4 sequence corresponds to the sequence of points (2n-3)P on the curve, where P = (0, 1). - Andrew Hone (anwh(AT)kent.ac.uk), Oct 12 2005
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MAPLE
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Digits:=11; f(x):=4*x^3-4*x+1; sols:=evalf(solve(f(x), x)); e1:=Re(sols[1]); e3:=Re(sols[2]); w1:=evalf(Int((f(x))^(-0.5), x=e1..infinity)); w3:=I*evalf(Int((-f(x))^(-0.5), x=-infinity..e3)); k:=2*w1-evalf(Int((f(x))^(-0.5), x=1..infinity)); z0:=w3+evalf(Int((f(x))^(-0.5), x=e3..-1)); A:=1/WeierstrassSigma(z0, 4.0, -1.0); B:=WeierstrassSigma(k, 4.0, -1.0)/WeierstrassSigma(z0+k, 4.0, -1.0)/A; for n from 0 to 10 do a[n]:=A*B^n*WeierstrassSigma(z0+n*k, 4.0, -1.0)/(WeierstrassSigma(k, 4.0, -1.0))^(n^2) od; (Andrew Hone (anwh(AT)kent.ac.uk), Oct 12 2005)
A006720 := proc(n)
option remember;
if n <= 3 then
1;
else
(procname(n-1)*procname(n-3)+procname(n-2)^2)/procname(n-4) ;
end if;
end proc: # R. J. Mathar, Jul 12 2012
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MATHEMATICA
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a[0] = a[1] = a[2] = a[3] = 1; a[n_] := a[n] = (a[n - 1] a[n - 3] + a[n - 2]^2)/a[n - 4]; Array[a, 23] (* Robert G. Wilson v, Jul 04 2007 *)
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PROG
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(PARI) a=vector(99); a[1]=a[2]=a[3]=a[4]=1; for(n=5, #a, a[n]=(a[n-1]*a[n-3]+a[n-2]^2)/a[n-4]); a \\ Charles R Greathouse IV, Jun 16 2011
(Haskell)
a006720 n = a006720_list !! n
a006720_list = [1, 1, 1, 1] ++
zipWith div (foldr1 (zipWith (+)) (map b [1..2])) a006720_list
where b i = zipWith (*) (drop i a006720_list) (drop (4-i) a006720_list)
-- Reinhard Zumkeller, Jan 22 2012
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CROSSREFS
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Cf. A006721, A006722, A006723, A048736, A028945, A028935, A151502.
For primes see A129739, A129740, A129741.
a(n)=(-1)^n*A006769(2n-3).
Cf. A165896. [From Jaume Oliver Lafont, Sep 29 2009]
Sequence in context: A037231 A082449 A129741 * A084710 A088173 A129739
Adjacent sequences: A006717 A006718 A006719 * A006721 A006722 A006723
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KEYWORD
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nonn,easy,nice
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AUTHOR
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N. J. A. Sloane.
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STATUS
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approved
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