A097495 Subsequence of terms of even index in the Somos-5 sequence.

1, 1, 1, 3, 11, 83, 1217, 22833, 1249441, 68570323, 11548470571, 2279343327171, 979023970244321, 771025645214210753, 816154448855663209121, 2437052403320731070558403, 7362326966302540624120605547

Offset: 0

Author

Andrew Hone, Aug 24 2004

Keywords

• nonn

Comments

The sequence corresponds to the sequence of points Q+nP on the curve y^2 = 4*x^3 - (121/12)*x + 845/216, where Q=(-19/12,2) and P=(17/12,-1).

For every 5th-order bilinear recurrence of Somos-5 type, b(n+3)*b(n-2) = alpha*b(n+2)*b(n-1) + beta*b(n+1)*b(n) (alpha, beta constant), both the subsequence of even index a(n)=b(2n) and that of odd index a(n)=b(2n+1) satisfy the same 4th-order Somos-4 type recurrence a(n+2)*a(n-2) = gamma*a(n+1)*a(n-1) + delta*a(n)^2, where the constant coefficients gamma, delta can be given in terms of alpha, beta and the initial data b(0), b(1), b(2), b(3), b(4).

a(n+2) is the Hankel transform of A174171. - Paul Barry, Mar 10 2010

This is a generalized Somos-4 sequence. - Michael Somos, May 12 2022

Links

• Paul Barry, Invariant number triangles, eigentriangles and Somos-4 sequences, arXiv preprint arXiv:1107.5490 [math.CO], 2011.
• A. N. W. Hone, Elliptic curves and quadratic recurrence sequences, Bull. Lond. Math. Soc. 37 (2005) 161-171.

Crossrefs

Cf. A006721, A006720.

Programs

Mathematica

RecurrenceTable[{a[0]==a[1]==a[2]==1,a[3]==3,a[n]==(a[n-1]a[n-3]+ 8a[n-2]^2)/a[n-4]},a,{n,20}] (* Harvey P. Dale, Sep 14 2013 *)
a[ n_] := a[n] = Which[n<1, a[2-n], n<4, {1, 1, 3}[[n]], True, (a[n-1]*a[n-3] + 8*a[n-2]^2)/a[n-4]]; (* Michael Somos, May 12 2022 *)

PARI

{a(n) = if(n<1, a(2-n), n<4, [1, 1, 3][n], (a(n-1)*a(n-3) + 8*a(n-2)^2)/a(n-4))}; /* Michael Somos, May 12 2022 */

Formula

a(n) = (a(n-1)*a(n-3) + 8*a(n-2)^2)/a(n-4).

Exact formula: a(n) = A*B^n*sigma(c+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,

A = 1/sigma(c) = 0.142427718 - 1.037985022*i,

B = sigma(k)*sigma(c)/sigma(c+k)

= 0.341936209 + 0.389300717*i,

c = Integral_{infinity..-19/12} dx/y

= 0.163392410 + 0.973928783*i,

k = Integral_{17/12..infinity} dx/y

= 1.018573545,

all to 9 decimal places.

a(n) = a(2-n) = (-8*a(n-1)*a(n-4) + 57*a(n-2)*a(n-3))/a(n-5) for all n in Z. - Michael Somos, May 12 2022