A105236 a(n+5) = (a(n+4)*a(n+1) + 2*a(n+3)*a(n+2))/a(n).

1, 1, 1, 1, 1, 3, 5, 11, 41, 233, 689, 5337, 49081, 458299, 3603685, 93208147, 1476087601, 27470407569, 816413467841, 43620306030449, 1172020019840081, 70063780891581107, 5804382690927311525, 511286588817798535899

Offset: 0

Author

Andrew Hone, Apr 14 2005

Keywords

• nonn

Comments

This is a bilinear recurrence of Somos 5 type, hence the terms a(n) are associated with a sequence of points P_n = P_0 + n*P on an elliptic curve E. In this case the curve E has integral j-invariant j=10976.

Links

• Seiichi Manyama, Table of n, a(n) for n = 0..147
• A. N. W. Hone, Sigma function solution of the initial value problem for Somos 5 sequences, arXiv:math/0501554 [math.NT], 2005-2006.
• A. N. W. Hone, Bilinear recurrences and addition formulas for hyperelliptic sigma functions, arXiv:math/0501162 [math.NT], 2005.
• A. N. W. Hone, Elliptic curves and quadratic recurrence sequences, Bull. Lond. Math. Soc. 37 (2005) 161-171.
• A. J. van der Poorten, Elliptic curves and continued fractions, J. Int. Sequences, Volume 8, no. 2 (2005), article 05.2.5.

Crossrefs

Cf. A006720, A006721.

Programs

Magma

[n le 5 select 1 else (Self(n-1)*Self(n-4) +2*Self(n-2)*Self(n-3))/Self(n-5): n in [1..41]]; // G. C. Greubel, Nov 26 2022

Mathematica

RecurrenceTable[{a[0]==a[1]==a[2]==a[3]==a[4]==1,a[n]==(2 a[-3+n] a[-2+n]+a[-4+n] a[-1+n])/a[-5+n]},a,{n,30}] (* Harvey P. Dale, Sep 15 2013 *)

SageMath

@CachedFunction
def a(n): # a = A105236
  if (n<5): return 1
  else: return (a(n-1)*a(n-4) +2*a(n-2)*a(n-3))/a(n-5)
[a(n) for n in range(41)] # G. C. Greubel, Nov 26 2022