A048736 -id:A048736 - cp's OEIS frontend

A006720 Somos-4 sequence: a(0)=a(1)=a(2)=a(3)=1; for n >= 4, a(n) = (a(n-1) * a(n-3) + a(n-2)^2) / a(n-4).

1, 1, 1, 1, 2, 3, 7, 23, 59, 314, 1529, 8209, 83313, 620297, 7869898, 126742987, 1687054711, 47301104551, 1123424582771, 32606721084786, 1662315215971057, 61958046554226593, 4257998884448335457, 334806306946199122193, 23385756731869683322514, 3416372868727801226636179

Offset: 0

N. J. A. Sloane

From the 5th term on, all terms have a primitive divisor; in other words, a prime divisor that divides no earlier term in the sequence. A proof appears in the Everest-McLaren-Ward paper. - Graham Everest (g.everest(AT)uea.ac.uk), Oct 26 2005
Twelve prime terms are known, occurring at indices 4, 5, 6, 7, 8, 11, 13, 16, 43, 52, 206, 647. The last two have been checked for probable primality only. The 647th term has 18498 decimal digits. Possibly these are the only prime terms in the entire sequence. - Graham Everest (g.everest(AT)uea.ac.uk), Nov 28 2006
The density of primes dividing some term in the sequence is 11/21. - Jeremy Rouse, Sep 18 2013
a(n) is a divisor of a(n+k*(2*n-3)) for all integers n and k. - Peter H van der Kamp, May 18 2015
a(n) is a divisor of A051138(k*(2*n-3)) for all integers n and k. - Helmut Ruhland, Jan 26 2024

Miklos Bona, editor, Handbook of Enumerative Combinatorics, CRC Press, 2015, page 565.
G. Everest, A. van der Poorten, I. Shparlinski and T. Ward, Recurrence Sequences, Amer. Math. Soc., 2003; pp. 9, 179.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Robert G. Wilson v, Table of a(n) for n = 0..100.
Paul Barry, Riordan-Bernstein Polynomials, Hankel Transforms and Somos Sequences, Journal of Integer Sequences, Vol. 15 2012, #12.8.2. - From _N. J. A. Sloane_, Dec 29 2012
Paul Barry, On the Hurwitz Transform of Sequences, Journal of Integer Sequences, Vol. 15 (2012), #12.8.7.
Paul Barry, Riordan arrays, the A-matrix, and Somos 4 sequences, arXiv:1912.01126 [math.CO], 2019.
Paul Barry, Integer sequences from elliptic curves, arXiv:2306.05025 [math.NT], 2023.
H. W. Braden, V. Z. Enolskii and A. N. W. Hone, Bilinear recurrences and addition formulas for hyperelliptic sigma functions, arXiv:math/0501162 [math.NT], 2005.
R. H. Buchholz and R. L. Rathbun, An infinite set of Heron triangles with two rational medians, Amer. Math. Monthly, 104 (1997), 107-115.
Xiangke Chang and Xingbiao Hu, A conjecture based on Somos-4 sequence and its extension, Linear Algebra Appl. 436, No. 11, 4285-4295 (2012).
Harini Desiraju and Brady Haran, The Troublemaker Number, Numberphile video (2022).
S. B. Ekhad and D. Zeilberger, How To Generate As Many Somos-Like Miracles as You Wish, arXiv preprint arXiv:1303.5306[math.CO], 2013.
Graham Everest, Gerard Mclaren and Tom Ward, Primitive divisors of elliptic divisibility sequences, arXiv:math/0409540 [math.NT], 2004-2006.
G. Everest, S. Stevens, D. Tamsett and T. Ward, Primitive divisors of quadratic polynomial sequences, arXiv:math/0412079v1 [math.NT], 2004.
G. Everest et al., Primes generated by recurrence sequences, arXiv:math/0412079 [math.NT], 2006.
G. Everest et al., Primes generated by recurrence sequences, Amer. Math. Monthly, 114 (No. 5, 2007), 417-431.
S. Fomin and A. Zelevinsky, The Laurent phenomenon, arXiv:math/0104241 [math.CO], 2001.
Allan Fordy and Andrew Hone, Discrete integrable systems and Poisson algebras from cluster maps, arXiv preprint arXiv:1207.6072 [nlin.SI], 2012.
A. P. Fordy, Periodic Cluster Mutations and Related Integrable Maps, arXiv preprint arXiv:1403.8061 [math-ph], 2014.
A. P. Fordy, Mutation-periodic quivers, integrable maps and associated Poisson algebras, Phil Trans. R. Soc. Lond. Ser A (Math. Phys. Eng. Sci.) 369 (1939) (2011) 1264-1279.
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], 2007.
A. N. W. Hone, Sigma function solution of the initial value problem for Somos 5 sequences, arXiv:math/0501554 [math.NT], 2005-2006.
A. N. W. Hone, Algebraic curves, integer sequences and a discrete Painlevé transcendent, Proceedings of SIDE 6, Helsinki, Finland, 2004; arXiv:0807.2538 [nlin.SI], 2008.
A. N. W. Hone, Elliptic curves and quadratic recurrence sequences, Bull. Lond. Math. Soc. 37 (2005) 161-171.
Andrew N. W. Hone, Growth of Mahler measure and algebraic entropy of dynamics with the Laurent property, arXiv:2109.08217 [math.NT], 2021.
A. N. W. Hone and R. Inoue, Discrete Painlevé equations from Y-systems, arXiv preprint arXiv:1405.5379 [math-ph], 2014
R. Jones and J. Rouse, Galois Theory of Iterated Endomorphisms, arXiv:0706.2384 [math.NT], 2007-2009; Proceedings of the London Mathematical Society, 100, no. 3 (2010), 763-794.
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.
Valentin Ovsienko and Serge Tabachnikov, Dual numbers, weighted quivers, and extended Somos and Gale-Robinson sequences, arXiv:1705.01623 [math.CO], 2017. See p. 3.
Kevin I. Piterman and Leandro Vendramin, Computer algebra with GAP, 2023. See p. 39.
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.
Helmut Ruhland, Somos-4 and a quartic Surface in RP^3, arXiv:2312.02085 [math.AG], 2023.
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.
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
Michael Somos, Four polynomial sequences w,x,y,z are discrete versions of the four Jacobi theta functions or the four Weierstrass sigma functions, 2016.
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, Somos-4 equation and related equations, arXiv:2307.05866 [math.CA], 2023.
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, 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.
A. J. van der Poorten, Elliptic curves and continued fractions, J. Int. Sequences, Volume 8, no. 2 (2005), article 05.2.5.
Leandro Vendramin, Mini-couse on GAP - Exercises, Universidad de Buenos Aires (Argentina, 2020).
Eric Weisstein's World of Mathematics, Somos Sequence
Index entries for two-way infinite sequences

Cf. A006721, A006722, A006723, A006769, A048736, A028945, A028935, A151502, A165896, A188313, A051138, A006769.
For primes see A129739, A129740, A129741.
Cf. A227199 (primes dividing some term).
Cf. A178384, A247368.

a006720 n = a006720_list !! n
a006720_list = [1,1,1,1] ++
   zipWith div (foldr1 (zipWith (+)) (map b [1..2])) a006720_list
   where b i = zipWith (*) (drop i a006720_list) (drop (4-i) a006720_list)
-- Reinhard Zumkeller, Jan 22 2012

I:=[1,1,1,1]; [n le 4 select I[n] else (Self(n-1)*Self(n-3)+Self(n-2)^2)/Self(n-4): n in [1..30]]; // Vincenzo Librandi, Aug 07 2017

Digits:=11; f(x):=4*x^3-4*x+1;sols:=evalf(solve(f(x),x)); e1:=Re(sols[1]); e3:=Re(sols[2]); w1:=evalf(Int((f(x))^(-0.5),x=e1..infinity)); w3:=I*evalf(Int((-f(x))^(-0.5),x=-infinity..e3)); k:=2*w1-evalf(Int((f(x))^(-0.5),x=1..infinity)); z0:=w3+evalf(Int((f(x))^(-0.5),x=e3..-1)); A:=1/WeierstrassSigma(z0,4.0,-1.0); B:=WeierstrassSigma(k,4.0,-1.0)/WeierstrassSigma(z0+k,4.0,-1.0)/A; for n from 0 to 10 do a[n]:=A*B^n*WeierstrassSigma(z0+n*k,4.0,-1.0)/(WeierstrassSigma(k,4.0,-1.0))^(n^2) od; # Andrew Hone, Oct 12 2005
A006720 := proc(n)
    option remember;
    if n <= 3 then
        1;
    else
        (procname(n-1)*procname(n-3)+procname(n-2)^2)/procname(n-4) ;
    end if;
end proc: # R. J. Mathar, Jul 12 2012

a[0] = a[1] = a[2] = a[3] = 1; a[n_] := a[n] = (a[n - 1] a[n - 3] + a[n - 2]^2)/a[n - 4]; Array[a, 23] (* Robert G. Wilson v, Jul 04 2007 *)
RecurrenceTable[{a[0]==a[1]==a[2]==a[3]==1,a[n]==(a[n-1]a[n-3]+a[n-2]^2)/ a[n-4]},a,{n,30}] (* Harvey P. Dale, Apr 07 2018 *)
b[ n_] := If[-2<=n<=2, {2, 1, 1, 3, 23}[[n+3]], 2*a[n+2]^3*a[n+3] + a[n+1]^2*(a[n+3]*a[n+4] - a[n+2]*a[n+5])]; a[ n_] := If[OddQ[n], b[(n-3)/2], b[-n/2]]; (* Michael Somos, Feb 28 2022 *)

a=vector(99);a[1]=a[2]=a[3]=a[4]=1;for(n=5,#a,a[n]=(a[n-1]*a[n-3]+a[n-2]^2)/a[n-4]); a \\ Charles R Greathouse IV, Jun 16 2011

from gmpy2 import divexact
A006720 = [1, 1, 1, 1]
for n in range(4, 101):
    A006720.append(divexact(A006720[n-1]*A006720[n-3]+A006720[n-2]**2,A006720[n-4]))
# Chai Wah Wu, Sep 01 2014

a(n) = a(3-n) = (-1)^n * A006769(2*n-3) for all n in Z.
a(n+1)/a(n) seems to be asymptotic to C^n with C = 1.226.... - Benoit Cloitre, Aug 07 2002. Confirmed by Hone - see below.
The terms of the sequence have the leading order asymptotics log a(n) ~ D n^2 with D = zeta(w1)*k^2/(2*w1) - log|sigma(k)| = 0.10222281... where zeta and sigma are the Weierstrass functions with invariants g2 = 4, g3 = -1, w1 = 1.496729323 is the real half-period of the corresponding elliptic curve, k = -1.134273216 as above. This agrees with Benoit Cloitre's numerical result with C = exp(2D) = 1.2268447... - Andrew Hone, Feb 09 2005
a(n) = (a(n-1)*a(n-3) + a(n-2)^2)/a(n-4); a(0) = a(1) = a(2) = a(3) = 1; exact formula is a(n) = A*B^n*sigma (z_0+nk)/(sigma (k))^(n^2), where sigma is the Weierstrass sigma function associated to the elliptic curve y^2 = 4*x^3-4*x+1, A = 1/sigma(z_0) = 0.112724016 - 0.824911687*i, B = sigma(k)*sigma (z_0)/sigma (z_0+k) = 0.215971963 + 0.616028193*i, k = 1.859185431, z_0 = 0.204680500 + 1.225694691*i, sigma(k) = 1.555836426, all to 9 decimal places. This is a special case of a general formula for 4th-order bilinear recurrences. The Somos-4 sequence corresponds to the sequence of points (2n-3)P on the curve, where P = (0, 1). - Andrew Hone, Oct 12 2005
a(2*n) = b(-n), a(2*n+1) = b(n-1) where b(n) = A188313(n) for all n in Z. - Michael Somos, Feb 27 2022

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.

Original entry on oeis.org

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

N. J. A. Sloane

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

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).

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

Cf. A006720, A006722, A006723, A048736.

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

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

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

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 *)

{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 */

{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 */

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

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.

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

Definition corrected by Chai Wah Wu, Aug 15 2014

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.

Original entry on oeis.org

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

N. J. A. Sloane

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).

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

Cf. A006720, A006721, A006723, A048736.

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

[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

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 *)

{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 */

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

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)

More terms from James A. Sellers, Aug 22 2000

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.

Original entry on oeis.org

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

N. J. A. Sloane

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

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

Cf. A006720, A006721, A006722, A048736.

Cf. A078495.

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

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

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 *)

{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])};

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

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

More terms from James A. Sellers, Aug 22 2000

A072881 a(1)=a(2)=a(3)=1; for n>3, a(n)=(a(n-1)*a(n-2)+a(n-1)+a(n-2))/a(n-3).

Original entry on oeis.org

1, 1, 1, 3, 7, 31, 85, 393, 1093, 5071, 14119, 65523, 182449, 846721, 2357713, 10941843, 30467815, 141397231, 393723877, 1827222153, 5087942581, 23612490751, 65749529671, 305135157603, 849655943137, 3943144558081

Offset: 1

Benoit Cloitre, Jul 28 2002, revised Feb 03 2005

What accounts for the high proportion of semiprimes in this sequence? Primes: 3, 7, 31, 1093, 846721, 393723877, ... Semiprimes: 85 = 5 * 17 393 = 3 * 131 5071 = 11 * 461 14119 = 7 * 2017 65523 = 3 * 21841 182449 = 43 * 4243 5087942581 = 11113 * 457837 849655943137 = 17 * 49979761361 3943144558081 = 31 * 127198211551 - Jonathan Vos Post, Feb 04 2005

Seiichi Manyama, Table of n, a(n) for n = 1..1803
P. Heideman and E. Hogan, A new family of Somos-like recurrences.
P. Heideman and E. Hogan, A new family of Somos-like recurrences, El. J. Combin. 15 (2008) #R54. [From _R. J. Mathar_, Dec 04 2008]
Index entries for linear recurrences with constant coefficients, signature (0, 14, 0, -14, 0, 1).

A048736 [From Jaume Oliver Lafont, Sep 25 2009]

Cf. A092264, A133846, A133847, A133848, A133854.

LinearRecurrence[{0, 14, 0, -14, 0, 1},{1, 1, 1, 3, 7, 31},26] (* Ray Chandler, Jul 24 2016 *)
nxt[{a_,b_,c_}]:={b,c,(c*b+c+b)/a}; NestList[nxt,{1,1,1},30][[All,1]] (* Harvey P. Dale, Mar 11 2019 *)

a(k=3, n) = {K = (k-1)/2; vds = vector(n); for (i=1, 2*K+1, vds[i] = 1;); for (i=2*K+2, n, vds[i] = (vds[i-1]*vds[i-2*K]+vds[i-K]+vds[i-K-1])/vds[i-2*K-1];); for (i=1, n, print1(vds[i], ","););} \\ Michel Marcus, Oct 28 2012

Both sequences u=(a(2n-1)){n>0} and u=(a(2n)){n>0} satisfy the order 3 linear recursion : u(n)=14u(n-1)-14u(n-2)+u(n-3).
a(2*n-1) = ceiling((1/11)*sqrt(1002/5-78*sqrt(33/5))*(sqrt(15)/2+sqrt(11)/ 2)^(2*n-1)).
a(2*n) = ceiling((1/11)*(13-sqrt(165))*(sqrt(15)/2+sqrt(11)/2)^(2*n)).
G.f.: x*(1+x-13*x^2-11*x^3+7*x^4+3*x^5)/(1-14*x^2+14*x^4-x^6). - Jaume Oliver Lafont, Sep 25 2009
a(n) = (4-(-1)^n)*a(n-1)-a(n-2)-1. - Bruno Langlois, Aug 21 2016
Sequences u=(a(2n)) and v=(a(2n-1)) satisfy order 2 linear recursions : u(n)=13*u(n-1)-u(n-2)-5 and v(n)=13*v(n-1)-v(n-2)-7. - Bruno Langlois, Aug 21 2016

A275173 a(n) = (a(n-3) + a(n-1) * a(n-5)) / a(n-6), a(0) = a(1) = ... = a(5) = 1.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 2, 3, 4, 6, 9, 22, 36, 51, 82, 129, 321, 529, 753, 1217, 1921, 4786, 7891, 11236, 18166, 28681, 71462, 117828, 167779, 271266, 428289, 1067137, 1759521, 2505441, 4050817, 6395649, 15935586, 26274979, 37413828, 60490982, 95506441, 237966646

Offset: 0

Seiichi Manyama, Jul 19 2016

Inspired by A048736.

Colin Barker, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (0,0,0,0,16,0,0,0,0,-16,0,0,0,0,1).

Cf. A101879, A048736, A275174.

Variants: A275175, A275176.

RecurrenceTable[{a[n] == (a[n - 3] + a[n - 1] a[n - 5])/a[n - 6], a[1] == 1, a[2] == 1, a[3] == 1, a[4] == 1, a[5] == 1, a[6] == 1}, a, {n, 42}] (* or *)
CoefficientList[Series[(1 + x + x^2 + x^3 + x^4 - 15 x^5 - 14 x^6 - 13 x^7 - 12 x^8 - 10 x^9 + 9 x^10 + 6 x^11 + 4 x^12 + 3 x^13 + 2 x^14)/((1 - x) (1 + x + x^2 + x^3 + x^4) (1 - 15 x^5 + x^10)), {x, 0, 41}], x] (* Michael De Vlieger, Jul 19 2016 *)
nxt[{a_,b_,c_,d_,e_,f_}]:={b,c,d,e,f,(d+f*b)/a}; NestList[nxt,{1,1,1,1,1,1},50][[;;,1]] (* Harvey P. Dale, Jan 06 2024 *)

Vec((1 +x +x^2 +x^3 +x^4 -15*x^5 -14*x^6 -13*x^7 -12*x^8 -10*x^9 +9*x^10 +6*x^11 +4*x^12 +3*x^13 +2*x^14) / ((1 -x)*(1 +x +x^2 +x^3 +x^4)*(1 -15*x^5 +x^10)) + O(x^50)) \\ Colin Barker, Jul 19 2016

def A(k, l, n)
  a = Array.new(k * 2, 1)
  ary = [1]
  while ary.size < n + 1
    break if (a[1] * a[-1] + a[k] * l) % a[0] > 0
    a = *a[1..-1], (a[1] * a[-1] + a[k] * l) / a[0]
    ary << a[0]
  end
  ary
end
def A275173(n)
  A(3, 1, n)
end

G.f.: (1 +x +x^2 +x^3 +x^4 -15*x^5 -14*x^6 -13*x^7 -12*x^8 -10*x^9 +9*x^10 +6*x^11 +4*x^12 +3*x^13 +2*x^14) / ((1 -x)*(1 +x +x^2 +x^3 +x^4)*(1 -15*x^5 +x^10)). - Colin Barker, Jul 19 2016

a(n) = 16*a(n-5) - 16*a(n-10) + a(n-15). - G. C. Greubel, Jul 20 2016

cp's OEIS Frontend

A192241 Primes in Dana Scott's sequence (A048736).

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A192242 Positions of primes within Dana Scott's sequence (A048736).

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A006720 Somos-4 sequence: a(0)=a(1)=a(2)=a(3)=1; for n >= 4, a(n) = (a(n-1) * a(n-3) + a(n-2)^2) / a(n-4).

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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.

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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.

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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.

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