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A006721 Somos-5 sequence: a(n) = (a(n-1) * a(n-4) + a(n-2) * a(n-3)) / a(n-5), with a(0) = a(1) = a(2) = a(3) = a(4) = 1.
(Formerly M0735)
28
1, 1, 1, 1, 1, 2, 3, 5, 11, 37, 83, 274, 1217, 6161, 22833, 165713, 1249441, 9434290, 68570323, 1013908933, 11548470571, 142844426789, 2279343327171, 57760865728994, 979023970244321, 23510036246274433, 771025645214210753 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,6

COMMENTS

Using the addition formula for the Weierstrass sigma function it is simple to prove that the subsequence of even terms of a Somos-5 type sequence satisfy a 4th order recurrence of Somos-4 type and similarly the odd subsequence satisfies the same 4th order recurrence. - Andrew Hone, Aug 24 2004

log(a(n)) ~ 0.071626946 * n^2. (Hone)

The Brown link article gives interesting information about related sequences including recurrences and numerical approximations.

The n-th term is a divisor of the (n+k*(2*n-4))-th term for all integers n and k. - Peter H van der Kamp, May 18 2015

REFERENCES

Matthew Christopher Russell, Using experimental mathematics to conjecture and prove theorems in the theory of partitions and commutative and non-commutative recurrences, PhD Dissertation, Mathematics Department, Rutgers University, May 2016; https://pdfs.semanticscholar.org/fdeb/e20954dacb7ec7a24afe2cf491b951c5a28d.pdf. Also (better) http://www.math.rutgers.edu/~zeilberg/Theses/MatthewRussellThesis.pdf

Ma, Xinrong. "Magic determinants of Somos sequences and theta functions." Discrete Mathematics 310.1 (2010): 1-5.

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

K. S. Brown, A Quasi-Periodic Sequence

R. H. Buchholz and R. L. Rathbun, An infinite set of Heron triangles with two rational medians, Amer. Math. Monthly, 104 (1997), 107-115.

Bryant Davis, Rebecca Kotsonis, Jeremy Rouse, The density of primes dividing a term in the Somos-5 sequence, arXiv:1507.05896 [math.NT], 2015.

S. Fomin and A. Zelevinsky, The Laurent phenomemon, arXiv:math/0104241 [math.CO], 2001.

David Gale, The strange and surprising saga of the Somos sequences, in Mathematical Entertainments, 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]

A. N. W. Hone, Elliptic curves and quadratic recurrence sequences, Bull. Lond. Math. Soc. 37 (2005) 161-171.

A. N. W. Hone, Sigma function solution of the initial value problem for Somos 5 sequences, arXiv:math/0501554 [math.NT], 2005-2006.

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

J. Propp, The Somos Sequence Site

J. Propp, The 2002 REACH tee-shirt

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, 2011

M. Somos, Somos 6 Sequence

M. Somos, Brief history of the Somos sequence problem

D. E. Speyer, Perfect matchings and the octahedral recurrence, arXiv:math/0402452 [math.CO], 2004.

P. H. van der Kamp, Somos-4 and Somos-5 are arithmetic divisibility sequences, arXiv:1505.00194 [Math.NT], 2015.

A. J. van der Poorten, Elliptic curves and continued fractions, arXiv:math/0403225 [math.NT], 2004.

A. J. van der Poorten, Elliptic curves and continued fractions, J. Int. Sequences, Volume 8, no. 2 (2005), article 05.2.5.

A. J. van der Poorten, Recurrence relations for elliptic sequences..., arXiv:math/0412293 [math.NT], 2004.

A. J. van der Poorten, Hyperelliptic curves, continued fractions and Somos sequences, arXiv:math/0608247 [math.NT], 2006.

Eric Weisstein's World of Mathematics, Somos Sequence.

D. Zagier, Problems posed at the St Andrews Colloquium, 1996

Index entries for two-way infinite sequences

FORMULA

Comments from Andrew Hone, Aug 24 2004: "Both the even terms b(n)=a(2n) and odd terms b(n)=a(2n+1) satisfy the fourth order recurrence b(n)=(b(n-1)*b(n-3)+8*b(n-2)^2)/b(n-4).

"Hence the general formula is a(2n)=A*B^n*sigma(c+n*k)/sigma(k)^(n^2), a(2n+1)=D*E^n*sigma(f+n*k)/sigma(k)^(n^2) where sigma is the Weierstrass sigma function associated to the elliptic curve y^2=4*x^3-(121/12)*x+845/216 (this is birationally equivalent to the minimal model V^2+U*V+6*V=U^3+7*U^2+12*U given by van der Poorten).

"The real/imaginary half-periods of the curve are w1=1.181965956, w3=0.973928783*I and the constants are A=0.142427718-1.037985022*I, B=0.341936209+0.389300717*I, c=0.163392411+w3, k=1.018573545, D=-0.363554228-0.803200610*I, E=0.644801269+0.734118205*I, f=c+k/2-w1 all to 9 decimal places."

a(4 - n) = a(n). a(n+2) * a(n-2) = 2 * a(n+1) * a(n-1) - a(n)^2 if n is even. a(n+2) * a(n-2) = 3 * a(n+1) * a(n-1) - a(n)^2 if n is odd.

MAPLE

for n from 0 to 4 do a[n]:= 1 od:

for n from 5 to 50 do a[n]:=(a[n-1] * a[n-4] + a[n-2] * a[n-3]) / a[n-5] od:

seq(a[i], i=0..50); # Robert Israel, May 19 2015

MATHEMATICA

a[0] = a[1] = a[2] = a[3] = a[4] = 1; a[n_] := a[n] = (a[n - 1] a[n - 4] + a[n - 2] a[n - 3])/a[n - 5]; Array[a, 27, 0] (* Robert G. Wilson v, Aug 15 2010 *)

a[ n_] := If[ Abs [n - 2] < 3, 1, If[ n < 0, a[4 - n], a[n] = (a[n - 1] a[n - 4] + a[n - 2] a[n - 3]) / a[n - 5]]] (* Michael Somos, Jul 15 2011 *)

RecurrenceTable[{a[0]==a[1]==a[2]==a[3]==a[4]==1, a[n]==(a[n-1]a[n-4]+ a[n-2]a[n-3])/a[n-5]}, a, {n, 30}] (* Harvey P. Dale, Dec 25 2011 *)

PROG

(PARI) {a(n) = if( abs(n-2) < 3, 1, if( n<0, a(4-n), (a(n-1) * a(n-4) + a(n-2) * a(n-3)) / a(n-5)))}; /* Michael Somos, Jul 15 2011 */

(Haskell)

a006721 n = a006721_list !! n

a006721_list = [1, 1, 1, 1, 1] ++

  zipWith div (foldr1 (zipWith (+)) (map b [1..2])) a006721_list

  where b i = zipWith (*) (drop i a006721_list) (drop (5-i) a006721_list)

-- Reinhard Zumkeller, Jan 22 2012

(Python)

from gmpy2 import divexact

A006721 = [1, 1, 1, 1, 1]

for n in range(5, 1001):

....A006721.append(divexact(A006721[n-1]*A006721[n-4]+A006721[n-2]*A006721[n-3], A006721[n-5])) # Chai Wah Wu, Aug 15 2014

(MAGMA) I:=[1, 1, 1, 1, 1]; [n le 5 select I[n] else  (Self(n-1) * Self(n-4) + Self(n-2) * Self(n-3)) div Self(n-5): n in [1..30]]; // Vincenzo Librandi, May 18 2015

CROSSREFS

Cf. A006720, A006722, A006723, A048736.

Sequence in context: A124561 A167604 A065510 * A111289 A255420 A127181

Adjacent sequences:  A006718 A006719 A006720 * A006722 A006723 A006724

KEYWORD

easy,nonn,nice

AUTHOR

N. J. A. Sloane

EXTENSIONS

a(26)-a(27) from Robert G. Wilson v, Aug 15 2010

Definition corrected by Chai Wah Wu, Aug 15 2014

STATUS

approved

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Last modified October 17 22:48 EDT 2018. Contains 316297 sequences. (Running on oeis4.)