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.

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

Offset: 0

Author

N. J. A. Sloane

Keywords

• easy
• nonn
• nice

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

The elliptic curve y^2 + xy = x^3 + x^2 - 2x (LMFDB label 102.a1) has infinite order point P = (2, 2) and 2-torsion point T = (0, 0). Define d(n) = a(n+2). The x and y coordinates of nP + T have denominators d(n)^2 and d(n)^3. - Michael Somos, Oct 29 2022

References

• Paul C. Kainen, Fibonacci in Somos-5 ..., Fib. Q., 60:4 (2022), 362-364.
• 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.
• Xiang-Ke Chang and Xing-Biao Hu, A conjecture based on Somos-4 sequence and its extension, Linear Algebra Appl. 436, No. 11, 4285-4295 (2012).
• Bryant Davis, Rebecca Kotsonis, and Jeremy Rouse, The density of primes dividing a term in the Somos-5 sequence, arXiv:1507.05896 [math.NT], 2015.
• Harini Desiraju and Brady Haran, The Troublemaker Number, Numberphile video (2022).
• 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, 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]
• J. W. E. Harrow and A. N. W. Hone, Casting more light in the shadows: dual Somos-5 sequences, arXiv:2409.00406 [nlin.SI], 2024. See p. 2.
• Andrew N. W. Hone, Elliptic curves and quadratic recurrence sequences, Bull. Lond. Math. Soc. 37 (2005) 161-171.
• Andrew N. W. Hone, Sigma function solution of the initial value problem for Somos 5 sequences, arXiv:math/0501554 [math.NT], 2005-2006.
• Andrew N. W. Hone, Heron triangles with two rational median and Somos-5 sequences, arXiv:2107.03197 [math.NT], 2022.
• Andrew N. W. Hone, Heron triangles and the hunt for unicorns, arXiv:2401.05581 [math.NT], 2024.
• LMFDB, Elliptic Curve 102.a1 (Cremona label 102a1)
• Xinrong Ma, Magic determinants of Somos sequences and theta functions, Discrete Mathematics 310.1 (2010): 1-5.
• 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.
• 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; see also.
• 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
• D. E. Speyer, Perfect matchings and the octahedral recurrence, arXiv:math/0402452 [math.CO], 2004.
• Alex Stone, The Astonishing Behavior of Recursive Sequences, Quanta Magazine, Nov 16 2023, 13 pages.
• Andrei K. Svinin, Volterra map and related recurrences, arXiv:2502.06908 [nlin.SI], 2025. See p. 27.
• Peter 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: : every Somos 4 is a Somos k, 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

Crossrefs

Cf. A006720, A006722, A006723, A048736.

Programs

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

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

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 *)

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 */

PARI

{a(n) = my(E = ellinit([1, 1, 0, -2, 0]), P = [2, 2], T = [0, 0]); if(n == 2, 1, n = abs(n-2); sqrtint(denominator(elladd(E, T, ellmul(E, P, n))[1])))}; /* Michael Somos, Oct 29 2022 */

Python

from gmpy2 import divexact
A006721 = [1,1,1,1,1]
for n in range(5,1001):
    A006721.append(int(divexact(A006721[n-1]*A006721[n-4]+A006721[n-2]*A006721[n-3], A006721[n-5]))) # Chai Wah Wu, Aug 15 2014

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.

Extensions

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

Definition corrected by Chai Wah Wu, Aug 15 2014