A006722 Somos-6 sequence: a(n) = (a(n-1) * a(n-5) + a(n-2) * a(n-4) + a(n-3)^2) / a(n-6), a(0) = ... = a(5) = 1.

1, 1, 1, 1, 1, 1, 3, 5, 9, 23, 75, 421, 1103, 5047, 41783, 281527, 2534423, 14161887, 232663909, 3988834875, 45788778247, 805144998681, 14980361322965, 620933643034787, 16379818848380849, 369622905371172929, 20278641689337631649, 995586066665500470689

Offset: 0

Author

N. J. A. Sloane

Keywords

• nonn
• easy
• nice

References

• C. Pickover, Mazes for the Mind, St. Martin's Press, NY, 1992, p. 350.
• N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Links

• T. D. Noe, Table of n, a(n) for n=0..100
• R. H. Buchholz and R. L. Rathbun, An infinite set of Heron triangles with two rational medians, Amer. Math. Monthly, 104 (1997), 107-115.
• Chang, Xiangke; Hu, Xingbiao, A conjecture based on Somos-4 sequence and its extension, Linear Algebra Appl. 436, No. 11, 4285-4295 (2012).
• Yuri N. Fedorov and Andrew N. W. Hone, Sigma-function solution to the general Somos-6 recurrence via hyperelliptic Prym varieties, arXiv:1512.00056 [nlin.SI], 2015.
• S. Fomin and A. Zelevinsky, The Laurent phenomenon, arXiv:math/0104241 [math.CO], 2001.
• David Gale, The strange and surprising saga of the Somos sequences, Math. Intelligencer 13(1) (1991), pp. 40-42.
• R. W. Gosper and Richard C. Schroeppel, Somos Sequence Near-Addition Formulas and Modular Theta Functions, arXiv:math/0703470 [math.NT], 2007.
• A. N. W. Hone, Analytic solutions and integrability for bilinear recurrences of order six, Appl. Anal. 89, no.4 (2010), 473-492.
• J. L. Malouf, An integer sequence from a rational recursion, Discr. Math. 110 (1992), 257-261.
• R. M. Robinson, Periodicity of Somos sequences, Proc. Amer. Math. Soc., 116 (1992), 613-619.
• Vladimir Shevelev and Peter J. C. Moses, On a sequence of polynomials with hypothetically integer coefficients, arXiv preprint arXiv:1112.5715 [math.NT], 2011.
• Michael Somos, Somos 6 Sequence
• Michael Somos, Brief history of the Somos sequence problem
• Alex Stone, The Astonishing Behavior of Recursive Sequences, Quanta Magazine, Nov 16 2023, 13 pages.
• A. van der Poorten, Hyperelliptic curves, continued fractions and Somos sequences, arXiv:math/0608247 [math.NT], 2006.
• Eric Weisstein's World of Mathematics, Somos Sequence.
• Index entries for two-way infinite sequences

Crossrefs

Cf. A006720, A006721, A006723, A048736.

Programs

Haskell

a006722 n = a006722_list !! n
a006722_list = [1,1,1,1,1,1] ++
  zipWith div (foldr1 (zipWith (+)) (map b [1..3])) a006722_list
  where b i = zipWith (*) (drop i a006722_list) (drop (6-i) a006722_list)
-- Reinhard Zumkeller, Jan 22 2012

Magma

[n le 6 select 1 else (Self(n-1)*Self(n-5)+Self(n-2)*Self(n-4)+ Self(n-3)^2)/Self(n-6): n in [1..30]]; // Vincenzo Librandi, Dec 02 2015

Mathematica

a[n_ /; 0 <= n <= 5] = 1; a[n_] := a[n] = (a[n-1]*a[n-5] + a[n-2]*a[n-4] + a[n-3]^2) / a[n-6]; Table[a[n], {n, 0, 25}] (* Jean-François Alcover, Nov 22 2013 *)
RecurrenceTable[{a[0]==a[1]==a[2]==a[3]==a[4]==a[5]==1,a[n]==(a[n-1]a[n-5]+ a[n-2]a[n-4]+a[n-3]^2)/a[n-6]},a,{n,30}] (* Harvey P. Dale, Dec 20 2014 *)

PARI

{a(n) = if( n>-1 && n<6, 1, if( n<0, a(5 - n), (a(n - 1) * a(n - 5) + a(n - 2) * a(n - 4) + a(n-3) * a(n-3)) / a(n - 6)))}; /* Michael Somos, Jan 30 2012 */

Python

from gmpy2 import divexact
A006722 = [1,1,1,1,1,1]
for n in range(6,101):
    A006722.append(divexact(A006722[n-1]*A006722[n-5]+A006722[n-2]*A006722[n-4]+A006722[n-3]**2,A006722[n-6]))
# Chai Wah Wu, Sep 01 2014

Formula

a(n) = a(5-n).

Michael Somos found an explicit formula for a(n) in 1993, which is not as widely known as it should be. The following is a quotation from the "Somos 6 sequence" document mentioned in the Links section: (Start)

This sequence is one of a large class of sequences of numbers that satisfy a non-linear recurrence relation depending on previous terms. It is also one of the class of sequences which can be computed from a theta series, hence I call them theta sequences. Here are the details:

Fix the following seven constants:

c1 = 0.875782749065950194217251...,

c2 = 1.084125925473763343779968...,

c3 = 0.114986002186402203509006...,

c4 = 0.077115634258697284328024...,

c5 = 1.180397390176742642553759...,

c6 = 1.508030831265086447098989..., and

c7 = 2.551548771413081602906643... .

Consider the doubly indexed series: f(x,y) = c1*c2^(x*y)*sum(k2, (-1)^k2*sum(k1, g(k1,k2,x,y))) , where g(k1,k2,x,y) = c3^(k1*k1)*c4^(k2*k2)*c5^(k1*k2)*cos(c6*k1*x+c7*k2*y) . Here both sums range over all integers.

Then the sequence defined by a(n) = f(n-2.5,n-2.5) is the Somos 6 sequence. I announced this in 1993. (End) - N. J. A. Sloane, Dec 06 2015

From Andrew Hone and Yuri Fedorov, Nov 27 2015: (Start)

The following is an exact formula for a(n):

a(n+3) = A*B^n*C^(n^2 -1)*sigma(v_0 + n*v) / sigma(v)^(n^2),

where

A = C / sigma(v_0),

B = A^(-1)*sigma(v) / sigma(v_0+v),

C = i/sqrt(20) (with i the imaginary unit),

sigma is the two-variable Kleinian sigma-function associated with the genus two curve X: y^2 = 4*x^5 - 233*x^4 + 1624*x^3 - 422*x^2 + 36*x - 1, and

v and v_0 are two-component vectors in the Jacobian of X, being the images under the Abel map of the divisors P_1+P_2 - 2*infinity, Q_1 + Q_2 - 2*infinity, respectively, where points P_j and Q_j on X are given by

P_1 = ( -8 + sqrt(65), 20*i*(129 -16*sqrt(65)) ),

P_2 = ( -8 - sqrt(65), 20*i*(129 +16*sqrt(65)) ),

Q_1 = ( 5 + 2*sqrt(6), 4*i*(71 +sqrt(6)) ),

Q_2 = ( 5 - 2*sqrt{6}, 4*i*(71 -sqrt(6)) ).

The Abel map is based at infinity and calculated with respect to the basis of holomorphic differentials dx/y, x dx/y.

Approximate values from Maple are A = 0.0619-0.0317*i, B = -0.0000973-0.0000158*i, v = (-.341*i, .477*i), v_0 = (-.379-.150*i, -.259+.576*i).

(End)

Extensions

More terms from James A. Sellers, Aug 22 2000