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A072877
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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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5
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1, 1, 1, 1, 2, 3, 19, 119, 65339, 67258454, 959259994615659593, 171965197021698738644442682357, 12959040525296547835480490169418622922155526267774117749963303914461
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OFFSET
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1,5
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COMMENTS
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A variation of a Somos-4 sequence with a(n-2)^4 in place of a(n-2)^2.
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LINKS
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Seiichi Manyama, Table of n, a(n) for n = 1..17
Joshua Alman, Cesar Cuenca, and Jiaoyang Huang, Laurent phenomenon sequences, Journal of Algebraic Combinatorics 43(3) (2015), pp. 589-633.
S. Fomin and A. Zelevinsky, The Laurent Phenomenon, Advances in Applied Mathematics 28 (2002), pp. 119-144.
David Gale, The strange and surprising saga of the Somos sequences, Math. Intelligencer 13(1) (1991), pp. 40-42.
Andrew N. W. Hone, Diophantine non-integrability of a third order recurrence with the Laurent property, arXiv:math/0601324 [math.NT], 2006.
Andrew N. W. Hone, Diophantine non-integrability of a third order recurrence with the Laurent property, J. Phys. A: Math. Gen. 39 (2006), pp. L171-L177.
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FORMULA
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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
From Jon E. Schoenfield, May 12 2019: (Start)
It appears that, for n >= 1,
a(n) = ceiling(e^(c0*x^n + d0/x^n)) if n is even,
ceiling(e^(c1*x^n + d1/x^n)) if n is odd,
where
x = sqrt(2 + sqrt(3)) = (sqrt(2) + sqrt(6))/2
c0 = 0.024915247166055931001426396817534982995670642690...
c1 = 0.029604794868229453467890216788323427656809346011...
d0 = -10.535089427608481105514469573411011428431309483956...
d1 = -2.856773870202800001336732759121362374871088274450...
(End)
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MAPLE
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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
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CROSSREFS
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Cf. A006720, A022405, A061292, A065918, A072878, A072879, A072880, A074394, A178768.
Sequence in context: A009178 A141508 A119344 * A201378 A241350 A032329
Adjacent sequences: A072874 A072875 A072876 * A072878 A072879 A072880
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KEYWORD
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easy,nonn
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AUTHOR
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Benoit Cloitre, Jul 28 2002
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EXTENSIONS
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Definition corrected by Matthew C. Russell, Apr 24 2012
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STATUS
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approved
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