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. (Formerly M2457)

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

0,7

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

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)

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 *)

Prog

(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 */
(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
(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
(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

Crossrefs

Cf. A006720, A006721, A006723, A048736.

Sequence in context: A241398 A262483 A083366 * A251413 A039774 A114001

Adjacent sequences: A006719 A006720 A006721 * A006723 A006724 A006725

Keyword

nonn,easy,nice

Author

N. J. A. Sloane

Extensions

More terms from James A. Sellers, Aug 22 2000

Status

approved