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
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
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
Kevin S. Brown, A Quasi-Periodic Sequence, MathPages.
Ralph H. Buchholz and Randall 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(11) (2012), 4285-4295.
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).
Sergey Fomin and Andrei 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), 40-42.
R. William Gosper and Richard C. Schroeppel, Somos Sequence Near-Addition Formulas and Modular Theta Functions, arXiv:math/0703470 [math.NT], 2007.
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.
Paul C. Kainen, Fibonacci in Somos-5 by Complexification, Fib. Q., 60:4 (2022), 362-364.
Xinrong Ma, Magic determinants of Somos sequences and theta functions, Disc. Math. 310(1) (2010), 1-5.
Janice L. Malouf, An integer sequence from a rational recursion, Disc. Math. 110 (1992), 257-261.
Jim Propp, The Somos Sequence Site, 2024.
Jim Propp, The 2002 REACH tee-shirt, 2002.
Raphael 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, Ph. D. Dissertation, Math. Dept., Rutgers Univ. (May 2016). See also SemanticScholar.
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, 2022.
Michael Somos, Brief history of the Somos sequence problem, 1990-1994.
David E. Speyer, Perfect matchings and the octahedral recurrence, arXiv:math/0402452 [math.CO], 2004.
Alex Stone, The Astonishing Behavior of Recursive Sequences, Quanta Mag. (Nov 16 2023), 13 pages.
Andrei K. Svinin, Volterra map and related recurrences, arXiv:2502.06908 [nlin.SI], 2025. See p. 27.
Andrei K. Svinin, An infinite family of homogeneous discrete equations with the Laurent property, arXiv:2604.13115 [nlin.SI], 2026.
Peter H. van der Kamp, Somos-4 and Somos-5 are arithmetic divisibility sequences, arXiv:1505.00194 [math.NT], 2015.
Alfred J. van der Poorten, Elliptic curves and continued fractions, arXiv:math/0403225 [math.NT], 2004.
Alfred J. van der Poorten, Elliptic curves and continued fractions, J. Int. Seq. 8(2) (2005), Article 05.2.5.
Alfred J. van der Poorten, Recurrence relations for elliptic sequences: : every Somos 4 is a Somos k, arXiv:math/0412293 [math.NT], 2004.
Alfred 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.
Don Zagier, Problems posed at the St Andrews Colloquium, 1996.
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.
Consider the recurrence b(n) = (b(n-1)*b(n-2)+1)/(b(n-1)^2*b(n-2)^2*b(n-3)), with b(0) = 2, b(1) = b(2) = 1, then b(n) = a(n+1)*a(n-1)/a(n)^2. Let s(n) be a sequence of rational numbers based on the generating function with the continued fraction expansion G(x) = 1/(1-x/(1-x/(1-x/(1-b(1)*x/(1-x/(1-b(2)*x/(1-...))))))), then the Hankel transform of s starting at s(0) gives us the Somos-5 sequence. - Thomas Scheuerle, Jan 05 2026
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 */
(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 */
(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(int(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
KEYWORD
easy,nonn,nice
AUTHOR
EXTENSIONS
a(26)-a(27) from Robert G. Wilson v, Aug 15 2010
Definition corrected by Chai Wah Wu, Aug 15 2014
STATUS
approved
