A006723 Somos-7 sequence: a(n) = (a(n-1) * a(n-6) + a(n-2) * a(n-5) + a(n-3) * a(n-4)) / a(n-7), a(0) = ... = a(6) = 1. (Formerly M2456)

1, 1, 1, 1, 1, 1, 1, 3, 5, 9, 17, 41, 137, 769, 1925, 7203, 34081, 227321, 1737001, 14736001, 63232441, 702617001, 8873580481, 122337693603, 1705473647525, 22511386506929, 251582370867257, 9254211194697641, 215321535159114017

Offset

0,8

References

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).

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.

J. L. Malouf, An integer sequence from a rational recursion, Discr. Math. 110 (1992), 257-261.

J. Propp, The Somos Sequence Site

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 7 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(6 - n) = a(n) for all n in Z.

a(n) = ((8-2*(-1)^n)*a(n-5)*a(n-3)-a(n-4)^2)/a(n-8). - Bruno Langlois, Aug 09 2016

Mathematica

RecurrenceTable[{a[0]==a[1]==a[2]==a[3]==a[4]==a[5]==a[6]==1, a[n] == (a[n-1]a[n-6]+a[n-2]a[n-5]+a[n-3]a[n-4])/a[n-7]}, a, {n, 30}] (* Harvey P. Dale, Jan 19 2012 *)

Prog

(PARI)  {a(n) = my(v); if( n<0, n = 6-n); if( n<7, 1, n++; v = vector(n, k, 1); for( k=8, n, v[k] = (v[k-1] * v[k-6] + v[k-2] * v[k-5] + v[k-3] * v[k-4]) / v[k-7]); v[n])};
(Haskell)
a006723 n = a006723_list !! n
a006723_list = [1, 1, 1, 1, 1, 1, 1] ++
  zipWith div (foldr1 (zipWith (+)) (map b [1..3])) a006723_list
  where b i = zipWith (*) (drop i a006723_list) (drop (7-i) a006723_list)
-- Reinhard Zumkeller, Jan 22 2012
(Python)
from gmpy2 import divexact
A006723 = [1, 1, 1, 1, 1, 1, 1]
for n in range(7, 101):
....A006723.append(divexact(A006723[n-1]*A006723[n-6]+A006723[n-2]*A006723[n-5]+A006723[n-3]*A006723[n-4], A006723[n-7]))
# Chai Wah Wu, Sep 01 2014
(Magma) I:=[1, 1, 1, 1, 1, 1, 1]; [n le 7 select I[n] else (Self(n-1)*Self(n-6) + Self(n-2)*Self(n-5) + Self(n-3)*Self(n-4))/Self(n-7): n in [1..30]]; // G. C. Greubel, Feb 21 2018

Crossrefs

Cf. A006720, A006721, A006722, A048736.

Cf. A078495.

Sequence in context: A297011 A337780 A178717 * A217097 A298590 A262451

Adjacent sequences: A006720 A006721 A006722 * A006724 A006725 A006726

Keyword

nonn,easy,nice

Author

N. J. A. Sloane

Extensions

More terms from James A. Sellers, Aug 22 2000

Status

approved