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a(1) = a(2) = a(3) = a(4) = 1; a(n) = (a(n-1)*a(n-3) + a(n-2)^4)/a(n-4).
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%I #54 Jun 01 2022 21:50:29

%S 1,1,1,1,2,3,19,119,65339,67258454,959259994615659593,

%T 171965197021698738644442682357,

%U 12959040525296547835480490169418622922155526267774117749963303914461

%N a(1) = a(2) = a(3) = a(4) = 1; a(n) = (a(n-1)*a(n-3) + a(n-2)^4)/a(n-4).

%C A variation of a Somos-4 sequence with a(n-2)^4 in place of a(n-2)^2.

%H Seiichi Manyama, <a href="/A072877/b072877.txt">Table of n, a(n) for n = 1..17</a>

%H Joshua Alman, Cesar Cuenca, and Jiaoyang Huang, <a href="https://doi.org/10.1007/s10801-015-0647-5">Laurent phenomenon sequences</a>, Journal of Algebraic Combinatorics 43(3) (2015), pp. 589-633.

%H S. Fomin and A. Zelevinsky, <a href="http://dx.doi.org/10.1006/aama.2001.0770">The Laurent Phenomenon</a>, Advances in Applied Mathematics 28 (2002), pp. 119-144.

%H David Gale, <a href="http://dx.doi.org/10.1007/BF03024070">The strange and surprising saga of the Somos sequences</a>, Math. Intelligencer 13(1) (1991), pp. 40-42.

%H Andrew N. W. Hone, <a href="https://arxiv.org/abs/math/0601324">Diophantine non-integrability of a third order recurrence with the Laurent property</a>, arXiv:math/0601324 [math.NT], 2006.

%H Andrew N. W. Hone, <a href="https://doi.org/10.1088/0305-4470/39/12/L01">Diophantine non-integrability of a third order recurrence with the Laurent property</a>, J. Phys. A: Math. Gen. 39 (2006), pp. L171-L177.

%F Lim_{n->infinity} (log(log(a(n))))/n = log(2+sqrt(3))/2 = A065918/2 or about 0.658. - _Andrew Hone_, Nov 15 2005; corrected by _Michel Marcus_, May 12 2019

%F From _Jon E. Schoenfield_, May 12 2019: (Start)

%F It appears that, for n >= 1,

%F a(n) = ceiling(e^(c0*x^n + d0/x^n)) if n is even,

%F ceiling(e^(c1*x^n + d1/x^n)) if n is odd,

%F where

%F x = sqrt(2 + sqrt(3)) = (sqrt(2) + sqrt(6))/2

%F c0 = 0.024915247166055931001426396817534982995670642690...

%F c1 = 0.029604794868229453467890216788323427656809346011...

%F d0 = -10.535089427608481105514469573411011428431309483956...

%F d1 = -2.856773870202800001336732759121362374871088274450...

%F (End)

%p L[0]:=0; L[1]:=0; L[2]:=0; L[3]:=0; for n from 0 to 4000 do L[n+4]:=evalf(ln(1+exp(L[n+3]+L[n+1]-4*L[n+2]))+4*L[n+2]-L[n]): od: for n from 3990 to 4000 do print(evalf(ln(L[n+4])/(n+4))): od: #Note: L[n] is log(a[n]) # _Andrew Hone_, Nov 15 2005

%t nxt[{a_,b_,c_,d_}]:={b,c,d,(d*b+c^4)/a}; NestList[nxt,{1,1,1,1},15][[All,1]] (* _Harvey P. Dale_, Jun 01 2022 *)

%Y Cf. A006720, A022405, A061292, A065918, A072878, A072879, A072880, A074394, A178768.

%K easy,nonn

%O 1,5

%A _Benoit Cloitre_, Jul 28 2002

%E Definition corrected by _Matthew C. Russell_, Apr 24 2012