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

%I A105236 #32 Dec 14 2022 12:34:21
%S A105236 1,1,1,1,1,3,5,11,41,233,689,5337,49081,458299,3603685,93208147,
%T A105236 1476087601,27470407569,816413467841,43620306030449,1172020019840081,
%U A105236 70063780891581107,5804382690927311525,511286588817798535899
%N A105236 a(n+5) = (a(n+4)*a(n+1) + 2*a(n+3)*a(n+2))/a(n).
%C A105236 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.
%H A105236 Seiichi Manyama, <a href="/A105236/b105236.txt">Table of n, a(n) for n = 0..147</a>
%H A105236 A. N. W. Hone, <a href="https://arxiv.org/abs/math/0501554">Sigma function solution of the initial value problem for Somos 5 sequences</a>, arXiv:math/0501554 [math.NT], 2005-2006.
%H A105236 A. N. W. Hone, <a href="https://arxiv.org/abs/math/0501162">Bilinear recurrences and addition formulas for hyperelliptic sigma functions</a>, arXiv:math/0501162 [math.NT], 2005.
%H A105236 A. N. W. Hone, <a href="https://doi.org/10.1112/S0024609304004163">Elliptic curves and quadratic recurrence sequences</a>, Bull. Lond. Math. Soc. 37 (2005) 161-171.
%H A105236 A. J. van der Poorten, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL8/Poorten/vdp40.html">Elliptic curves and continued fractions</a>, J. Int. Sequences, Volume 8, no. 2 (2005), article 05.2.5.
%t A105236 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 *)
%o A105236 (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
%o A105236 (SageMath)
%o A105236 @CachedFunction
%o A105236 def a(n): # a = A105236
%o A105236   if (n<5): return 1
%o A105236   else: return (a(n-1)*a(n-4) +2*a(n-2)*a(n-3))/a(n-5)
%o A105236 [a(n) for n in range(41)] # _G. C. Greubel_, Nov 26 2022
%Y A105236 Cf. A006720, A006721.
%K A105236 nonn
%O A105236 0,6
%A A105236 _Andrew Hone_, Apr 14 2005