A072877 a(1) = a(2) = a(3) = a(4) = 1; a(n) = (a(n-1)*a(n-3) + a(n-2)^4)/a(n-4).

1, 1, 1, 1, 2, 3, 19, 119, 65339, 67258454, 959259994615659593, 171965197021698738644442682357, 12959040525296547835480490169418622922155526267774117749963303914461

Offset

1,5

Comments

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

Links

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.

Formula

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)

Maple

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

Mathematica

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 *)

Crossrefs

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

Sequence in context: A141508 A366357 A119344 * A201378 A241350 A032329

Adjacent sequences: A072874 A072875 A072876 * A072878 A072879 A072880

Keyword

easy,nonn

Author

Benoit Cloitre, Jul 28 2002

Extensions

Definition corrected by Matthew C. Russell, Apr 24 2012

Status

approved