0,6

Using the addition formula for the Weierstrass sigma function it is simple to prove that the subsequence of even terms of a Somos-5 type sequence satisfy a 4th order recurrence of Somos-4 type, and similarly the odd subsequence satsifies the same 4th order recurrence. - Andrew Hone (anwh(AT)kent.ac.uk), Aug 24 2004

log(a(n)) ~ 0.071626946 * n^2. (Hone)

R. H. Buchholz and R. L. Rathbun, "An infinite set of Heron triangles with two rational medians", Amer. Math. Monthly, 104 (1997), 107-115.

David Gale, "The strange and surprising saga of the Somos sequence", Math. Intelligencer 13(1) (1991), pp. 40-42.

A. N. W. Hone, Elliptic curves and quadratic recurrence sequences, Bull. Lond. Math. Soc. 37 (2005) 161-171.

J. L. Malouf, "An integer sequence from a rational recursion", Discr. Math. 110 (1992), 257-261.

Alfred J. van der Poorten, Elliptic Curves and Continued Fractions, Journal of Integer Sequences, Vol. 8 (2005), Article 05.2.5.

R. M. Robinson, "Periodicity of Somos sequences", Proc. Amer. Math. Soc., 116 (1992), 613-619.

T. D. Noe, Table of n, a(n) for n=0..100

S. Fomin and A. Zelevinsky, The Laurent phenomemon

A. N. W. Hone, Sigma function solution of the initial value problem for Somos 5 sequences

J. Propp, The Somos Sequence Site

J. Propp, The 2002 REACH tee-shirt

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, Elliptic curves and continued fractions

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, Link to a section of The World of Mathematics.

D. Zagier, Problems posed at the St Andrews Colloquium, 1996

Comments from Andrew Hone (anwh(AT)kent.ac.uk), Aug 24 2004: "Both the even terms b(n)=a(2n) and odd terms b(n)=a(2n+1) satisfy the fourth order recurrence b(n)=(b(n-1)*b(n-3)+8*b(n-2)^2)/b(n-4).

"Hence the general formula is a(2n)=A*B^n*sigma(c+n*k)/sigma(k)^(n^2), a(2n+1)=D*E^n*sigma(f+n*k)/sigma(k)^(n^2) where sigma is the Weierstrass sigma function associated to the elliptic curve y^2=4*x^3-(121/12)*x+845/216 (this is birationally equivalent to the minimal model V^2+U*V+6*V=U^3+7*U^2+12*U given by van der Poorten).

"The real/imaginary half-periods of the curve are w1=1.181965956, w3=0.973928783*I, and the constants are A=0.142427718-1.037985022*I, B=0.341936209+0.389300717*I, c=0.163392411+w3, k=1.018573545, D=-0.363554228-0.803200610*I, E=0.644801269+0.734118205*I, f=c+k/2-w1 all to 9 decimal places."

Cf. A006720, A006722, A006723, A048736.

Cf. A006720.

Sequence in context: A067078 A124561 A065510 this_sequence A111289 A127181 A113734

Adjacent sequences: A006718 A006719 A006720 this_sequence A006722 A006723 A006724

easy,nonn,nice

njas