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A097495 Subsequence of terms of even index in the Somos-5 sequence. 1
1, 1, 1, 3, 11, 83, 1217, 22833, 1249441, 68570323, 11548470571, 2279343327171, 979023970244321, 771025645214210753, 816154448855663209121, 2437052403320731070558403, 7362326966302540624120605547 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,4

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

REFERENCES

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

LINKS

Table of n, a(n) for n=0..16.

P. Barry, Invariant number triangles, eigentriangles and Somos-4 sequences, arXiv preprint arXiv:1107.5490, 2011

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.

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 *)

CROSSREFS

Cf. A006721, A006720.

Sequence in context: A342058 A062580 A335968 * A228034 A157980 A092148

Adjacent sequences:  A097492 A097493 A097494 * A097496 A097497 A097498

KEYWORD

nonn

AUTHOR

Andrew Hone, Aug 24 2004

STATUS

approved

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Last modified January 26 13:09 EST 2022. Contains 350598 sequences. (Running on oeis4.)