A006722 -id:A006722 - cp's OEIS frontend

A038754 a(2n) = 3^n, a(2n+1) = 2*3^n.

1, 2, 3, 6, 9, 18, 27, 54, 81, 162, 243, 486, 729, 1458, 2187, 4374, 6561, 13122, 19683, 39366, 59049, 118098, 177147, 354294, 531441, 1062882, 1594323, 3188646, 4782969, 9565938, 14348907, 28697814, 43046721, 86093442, 129140163, 258280326, 387420489

Offset: 0

Henry Bottomley, May 03 2000

In general, for the recurrence a(n) = a(n-1)*a(n-2)/a(n-3), all terms are integers iff a(0) divides a(2) and first three terms are positive integers, since a(2n+k) = a(k)*(a(2)/a(0))^n for all nonnegative integers n and k.
Equals eigensequence of triangle A070909; (1, 1, 2, 3, 6, 9, 18, ...) shifts to the left with multiplication by triangle A070909. - Gary W. Adamson, May 15 2010
The a(n) represent all paths of length (n+1), n >= 0, starting at the initial node on the path graph P_5, see the second Maple program. - Johannes W. Meijer, May 29 2010
a(n) is the difference between numbers of multiple of 3 evil (A001969) and odious (A000069) numbers in interval [0, 2^(n+1)). - Vladimir Shevelev, May 16 2012
A "half-geometric progression": to obtain a term (beginning with the third one) we multiply the before previous one by 3. - Vladimir Shevelev, May 21 2012
Pisano periods: 1, 2, 1, 4, 8, 2, 12, 4, 1, 8, 10, 4, 6, 12, 8, 8, 32, 2, 36, 8, ... . - R. J. Mathar, Aug 10 2012
Numbers k such that the k-th cyclotomic polynomial has a root mod 3. - Eric M. Schmidt, Jul 31 2013
Range of row n of the circular Pascal array of order 6. - Shaun V. Ault, Jun 05 2014
Also, the number of walks of length n on the graph 0--1--2--3--4 starting at vertex 1. - Sean A. Irvine, Jun 03 2025

			In the interval [0,2^5) we have 11 multiples of 3 numbers, from which 10 are evil and only one (21) is odious. Thus a(4) = 10 - 1 = 9. - _Vladimir Shevelev_, May 16 2012

Indranil Ghosh, Table of n, a(n) for n = 0..1500 (first 401 terms from T. D. Noe)
S. V. Ault and C. Kicey, Counting paths in corridors using circular Pascal arrays, Discrete Mathematics, Vol. 332 (2014), pp. 45-54.
Paul Barry, Three Études on a sequence transformation pipeline, arXiv:1803.06408 [math.CO], 2018.
Nachum Dershowitz, Between Broadway and the Hudson, arXiv:2006.06516 [math.CO], 2020.
Sean A. Irvine, Walks on Graphs.
Richard L. Ollerton and Anthony G. Shannon, Some properties of generalized Pascal squares and triangles, Fib. Q., 36 (1998), 98-109. See pages 106-7.
Vladimir Shevelev, On monotonic strengthening of Newman-like phenomenon on (2m+1)-multiples in base 2m, arXiv:0710.3177 [math.NT], 2007.
M. B. Wells, Elements of Combinatorial Computing, Pergamon, Oxford, 1971. [Annotated scanned copy of pages 237-240]
Index entries for linear recurrences with constant coefficients, signature (0,3).

Cf. A000045, A000069, A000079, A000120, A000244, A001969, A006720, A006721.
Cf. A006722, A006723, A008776, A028495, A030436, A037124, A048328, A061551.
Cf. A068911, A070909, A087503, A124302, A128588, A133464, A140730, A140740.
Cf. A178381, A182751, A182752, A182753, A182754, A182755, A182756, A182757.
Fifth row of the array A094718.

import Data.List (transpose)
a038754 n = a038754_list !! n
a038754_list = concat $ transpose [a000244_list, a008776_list]
-- Reinhard Zumkeller, Oct 19 2015

[n le 2 select n else 3*Self(n-2): n in [1..40]]; // Vincenzo Librandi, Aug 18 2016

a[0]:=0:a[1]:=1:for n from 2 to 50 do a[n]:=3*a[n-2]+2 od: seq(a[n]+1, n=0..34); # Zerinvary Lajos, Mar 20 2008
with(GraphTheory): P:=5: G:=PathGraph(P): A:= AdjacencyMatrix(G): nmax:=35; for n from 1 to nmax do B(n):=A^n; a(n):=add(B(n)[1,k],k=1..P) od: seq(a(n),n=1..nmax); # Johannes W. Meijer, May 29 2010

LinearRecurrence[{0,3},{1,2},40] (* Harvey P. Dale, Jan 26 2014 *)
CoefficientList[Series[(1+2x)/(1-3x^2), {x, 0, 40}], x] (* Vincenzo Librandi, Aug 18 2016 *)
Module[{nn=20,c},c=3^Range[0,nn];Riffle[c,2c]] (* Harvey P. Dale, Aug 21 2021 *)

PARI
```
a(n)=(1/6)*(5-(-1)^n)*3^floor(n/2)
```
PARI
```
a(n)=3^(n>>1)<
				
```

[2^(n%2)*3^((n-(n%2))/2) for n in range(61)] # G. C. Greubel, Oct 10 2022

a(n) = a(n-1)*a(n-2)/a(n-3) with a(0)=1, a(1)=2, a(2)=3.
a(2*n) = (3/2)*a(2*n-1) = 3^n, a(2*n+1) = 2*a(2*n) = 2*3^n.
From Benoit Cloitre, Apr 27 2003: (Start)
a(1)=1, a(n)= 2*a(n-1) if a(n-1) is odd, or a(n)= (3/2)*a(n-1) if a(n-1) is even.
a(n) = (1/6)*(5-(-1)^n)*3^floor(n/2).
a(2*n) = a(2*n-1) + a(2*n-2) + a(2*n-3).
a(2*n+1) = a(2*n) + a(2*n-1). (End)
G.f.: (1+2*x)/(1-3*x^2). - Paul Barry, Aug 25 2003
From Reinhard Zumkeller, Sep 11 2003: (Start)
a(n) = (1 + n mod 2) * 3^floor(n/2).
a(n) = A087503(n) - A087503(n-1). (End)
a(n) = sqrt(3)*(2+sqrt(3))*(sqrt(3))^n/6 - sqrt(3)*(2-sqrt(3))*(-sqrt(3))^n/6. - Paul Barry, Sep 16 2003
From Reinhard Zumkeller, May 26 2008: (Start)
a(n) = A140740(n+2,2).
a(n+1) = a(n) + a(n - n mod 2). (End)
If p(i) = Fibonacci(i-3) and if A is the Hessenberg matrix of order n defined by A(i,j) = p(j-i+1), (i<=j), A(i,j)=-1, (i=j+1), and A(i,j)=0 otherwise. Then, for n>=1, a(n-1) = (-1)^n det A. - Milan Janjic, May 08 2010
a(n) = A182751(n) for n >= 2. - Jaroslav Krizek, Nov 27 2010
a(n) = Sum_{i=0..2^(n+1), i==0 (mod 3)} (-1)^A000120(i). - Vladimir Shevelev, May 16 2012
a(0)=1, a(1)=2, for n>=3, a(n)=3*a(n-2). - Vladimir Shevelev, May 21 2012
Sum_(n>=0) 1/a(n) = 9/4. - Alexander R. Povolotsky, Aug 24 2012
a(n) = sqrt(3*a(n-1)^2 + (-3)^(n-1)). - Richard R. Forberg, Sep 04 2013
a(n) = 2^((1-(-1)^n)/2)*3^((2*n-1+(-1)^n)/4). - Luce ETIENNE, Aug 11 2014
From Reinhard Zumkeller, Oct 19 2015: (Start)
a(2*n) = A000244(n), a(2*n+1) = A008776(n).
For n > 0: a(n+1) = a(n) + if a(n) odd then min{a(n), a(n-1)} else max{a(n), a(n-1)}, see also A128588. (End)
E.g.f.: (7*cosh(sqrt(3)*x) + 4*sqrt(3)*sinh(sqrt(3)*x) - 4)/3. - Stefano Spezia, Feb 17 2022
Sum_{n>=0} (-1)^n/a(n) = 3/4. - Amiram Eldar, Dec 02 2022

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).

Original entry on oeis.org

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

A048736 Dana Scott's sequence: a(n) = (a(n-2) + a(n-1) * a(n-3)) / a(n-4), a(0) = a(1) = a(2) = a(3) = 1.

Original entry on oeis.org

1, 1, 1, 1, 2, 3, 5, 13, 22, 41, 111, 191, 361, 982, 1693, 3205, 8723, 15042, 28481, 77521, 133681, 253121, 688962, 1188083, 2249605, 6123133, 10559062, 19993321, 54419231, 93843471, 177690281, 483649942, 834032173, 1579219205, 4298430243, 7412446082, 14035282561, 38202222241, 65877982561

Offset: 0

David Johnson-Davies

The recursion has the Laurent property. If a(0), a(1), a(2), a(3) are variables, then a(n) is a Laurent polynomial (a rational function with a monic monomial denominator). - Michael Somos, Feb 05 2012

A generalization is if the recursion is modified to a(n) = (a(n-2) + a(n-1) * b*a(n-3)) / a(n-4) where b is a constant, and with arbitrary nonzero initial values, (a(0), a(1), a(2), a(3)), then a(n) = c*(a(n-3) - a(n-6)) + a(n-9) for all n in Z where c is another constant. - Michael Somos, Oct 28 2021

			G.f. = 1 + x + x^2 + x^3 + 2*x^4 + 3*x^5 + 13*x^6 + 22*x^7 + 41*x^8 + 111*x^9 + ...

Seiichi Manyama, Table of n, a(n) for n = 0..3165 (first 501 terms from T. D. Noe)
Joshua Alman, Cesar Cuenca, and Jiaoyang Huang, Laurent phenomenon sequences, Journal of Algebraic Combinatorics 43(3) (2015), 589-633.
Hal Canary, The Dana Scott Recurrence [From _Jaume Oliver Lafont_, Sep 25 2009]
S. Fomin and A. Zelevinsky, The Laurent phenomenon, arXiv:math/0104241 [math.CO], 2001.
S. Fomin and A. Zelevinsky, The Laurent Phenomenon, Advances in Applied Mathematics, 28 (2002), 119-144.
David Gale, The strange and surprising saga of the Somos sequences, Math. Intelligencer 13(1) (1991), pp. 40-42.
D. Gale, Tracking the Automatic Ant And Other Mathematical Explorations, A Collection of Mathematical Entertainments Columns from The Mathematical Intelligencer, Springer, 1998, p. 4.
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.
Eric Weisstein's World of Mathematics, Laurent Polynomial
Index entries for two-way infinite sequences
Index entries for linear recurrences with constant coefficients, signature (0,0,10,0,0,-10,0,0,1)

Cf. A006720, A006721, A006722, A006723, A092420, A072881.
Cf. A192241, A192242 (primes and where they occur).
Cf. A276531.

a048736 n = a048736_list !! n
a048736_list = 1 : 1 : 1 : 1 :
   zipWith div
     (zipWith (+)
       (zipWith (*) (drop 3 a048736_list)
                    (drop 1 a048736_list))
       (drop 2 a048736_list))
     a048736_list
-- Reinhard Zumkeller, Jun 26 2011

I:=[1,1,1,1]; [n le 4 select I[n] else (Self(n-2) + Self(n-1)*Self(n-3)) / Self(n-4): n in [1..30]]; // G. C. Greubel, Feb 20 2018

P:=proc(q) local n,v; v:=[1,1,1,1]; for n from 1 to q do
v:=[op(v),(v[-2]+v[-1]*v[-3])/v[-4]] od: op(v); end: P(35); # Paolo P. Lava, Aug 24 2025

RecurrenceTable[{a[0] == a[1] == a[2] == a[3] == 1, a[n] == (a[n - 2] + a[n - 1]a[n - 3])/a[n - 4]}, a[n], {n, 40}] (* or *) LinearRecurrence[{0, 0, 10, 0, 0, -10, 0, 0, 1}, {1, 1, 1, 1, 2, 3, 5, 13, 22}, 41] (* Harvey P. Dale, Oct 22 2011 *)

Vec((1+x+x^2-9*x^3-8*x^4-7*x^5+5*x^6+3*x^7+2*x^8) / (1-10*x^3+10*x^6-x^9)+O(x^99)) \\ Charles R Greathouse IV, Jul 01 2011

a(n) = 9*a(n-3) - a(n-6) - 3 - ( ceiling(n/3) - floor(n/3) ), with a(0) = a(1) = a(2) = a(3) = 1, a(4) = 2, a(5) = 3. - Michael Somos
From Jaume Oliver Lafont, Sep 17 2009: (Start)
a(n) = 10*a(n-3) - 10*a(n-6) + a(n-9).
G.f.: (1 + x + x^2 - 9*x^3 - 8*x^4 - 7*x^5 + 5*x^6 + 3*x^7 + 2*x^8)/(1 - 10*x^3 + 10*x^6 - x^9). (End)
a(n) = a(3-n) for all n in Z. - Michael Somos, Feb 05 2012

More terms from Michael Somos

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

A102276 a(n) = (a(n-1) * a(n-5) + a(n-3)^2) / a(n-6) with a(0) = ... = a(5) = 1, a(n) = a(5-n) for all n in Z.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 2, 3, 4, 8, 17, 50, 107, 239, 1103, 3775, 14463, 55283, 256666, 2059753, 9820288, 55075036, 503857819, 4083736906, 44590046729, 335845998321, 3581731774609, 68868876045617, 782035904796497, 11680434156713849, 194342679446776442

Offset: 0

Michael Somos, Jan 02 2005

nonn

Sequence defined by recursion derived from a genus 2 curve.
Similar to the Somos-6 and Somos-7 sequences with many bilinear identities.
If a0 := a(n), a1 := a(n+1), ..., a5 := a(n+5), a6 := a(n+6) and a6 = (a5*a1 + a3^2)/a0 for all n in Z, then c := (a0^2*a1*a4*a5^2 + a0^2*a3*a4^3 + a1^3*a2*a5^2 + a0*a2^2*a3*a4^2 + a1^2*a2*a3^2*a5 + a0*a2*a3^3*a4 + a1*a2^3*a3*a5 + a2^3*a3^3)/(a0*a1*a2*a3*a4*a5) is constant. - Michael Somos, Jun 30 2024

Seiichi Manyama, Table of n, a(n) for n = 0..211
A. J. van der Poorten, Curves of Genus 2, Continued Fractions and Somos Sequences, arXiv:math/0412372 [math.NT], 2004.
A. J. van der Poorten, Curves of Genus 2, Continued Fractions and Somos Sequences, J. Integer Seqs., 8 (2005), #05.3.4.
Index entries for two-way infinite sequences

Cf. A006720, A006722, A256858, A256916, A018896, A271341, A271835, A271831, A271837, A271838, A271839.

I:=[1, 2, 3, 4, 8, 17]; [1, 1, 1, 1, 1] cat [n le 6 select I[n] else (Self(n-1)*Self(n-5) + Self(n-3)^2)/Self(n-6): n in [1..30]]; // G. C. Greubel, Aug 03 2018

Join[{1, 1, 1, 1, 1}, RecurrenceTable[{a[n] == (a[n-1]*a[n-5] + a[n-3]^2)/a[n-6], a[6] == 1, a[7] == 2, a[8] == 3, a[9] == 4, a[10] == 8, a[11] == 17}, a, {n, 6, 60}]] (* G. C. Greubel, Aug 03 2018 *)

{a(n) = my(an); if( n<0, a(5-n), n++; an = vector(n,i,1); for(k=7, n, an[k] = (an[k-1]*an[k-5] + an[k-3]^2) / an[k-6]); an[n])};

a(n) = A256858(2*n - 5) for all n in Z. - Michael Somos, Apr 13 2015
Let b(n) = A256916(n). Then 0 = a(n) * b(n) - a(n-2) * b(n+2) + a(n-3) * b(n+3) for all n in Z. - Michael Somos, Apr 13 2015
0 = a(n) * a(n+6) - a(n+1) * a(n+5) - a(n+3) * a(n+3) for all n in Z. - Michael Somos, Apr 13 2015
0 = a(n) * a(n+9) + a(n+2) * a(n+7) - a(n+3) * a(n+6) - 9 * a(n+4) * a(n+5) for all n in Z. - Michael Somos, Apr 13 2015

A368482 The degree of polynomials related to Somos-6 sequences. Also for n > 4 the degree of the (n-5)-th involution in a family of involutions in the Cremona group of rank 5 defined by a Somos-6 sequence.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 2, 3, 4, 6, 8, 11, 13, 16, 20, 23, 27, 31, 36, 41, 45, 51, 57, 63, 69, 75, 83, 90, 97, 105, 113, 122, 130, 139, 149, 158, 168, 178, 189, 200, 210, 222, 234, 246, 258, 270, 284, 297, 310, 324, 338, 353, 367, 382, 398, 413, 429, 445, 462

Offset: 0

Helmut Ruhland, Dec 26 2023

nonn

Let s(0), s(1), ..., s(4), s(5) be the 6 initial values in a Somos-6 sequence. The following terms s(6), s(7), ... are rational expressions in the 6 initial values derived from the Somos-6 recurrence: s(n) = ( s(n-1)*s(n-5) + s(n-2)*s(n-4) + s(n-3)^2 ) / s(n-6). E.g., s(6) = (s(1)*s(5) + s(2)*s(4) + s(3)^2) / s(0), s(7) = ... .
Because of the Laurent property of a Somos-6 sequence the denominator of these terms is a monomial in the initial values.
With the sequence e(n) = A369113(n), the tropical version of the Somos-6 sequence, the monomial D(n) is defined as Product_{k=0..5} s(k)^a(n-k). Define the polynomial G(n) to be s(n) * D(n). G(n) is 1 for n < 4, else G(n) is the numerator of s(n), so ..., G(4) = 1, G(5) = 1, G(6) = s(1)*s(5) + s(2)*s(4) + s(3)^2, ...
For n >= 0, a term a(n) of the actual sequence is the degree of G(n). The degree of the denominator of s(n) is a(n) - 1.
This Somos-6 sequence defines a family (proposed name: Somos family) S of (birational) involutions in Cr_5(R), the Cremona group of rank 5.
A Somos involution S(n) in this family is defined as S(n) : RP^5 -> RP^5, (s(0) : s(1) : ... : s(4) : s(5)) -> (s(n+5) : s(n+4) : ... : s(n+1) : s(n)). For n > 0 S(n) = (G(n+5) : G(n+4)*m1 : ... : G(n+1)*m4 : G(n)*m5 ), with m1, m2, ..., m4, m5 monomials. The involutions generate an infinite dihedral group. Already 2 consecutive involutions S(n), S(n+1) generate this group too. This group as a dihedral group has 2 conjugacy classes { ..., S(0), S(2), S(4), ... } and { ..., S(1), S(3), S(5), ... } of involutions. The degree of such an involution S(n) equals the degree of G(n+5) and the term a(n+5) in the actual sequence.

S. Fomin and A. Zelevinsky, The Laurent phenomenon, arXiv:math/0104241 [math.CO], 2001.
Index entries for linear recurrences with constant coefficients, signature (2,-2,2,-1,1,-2,2,-2,1).

Cf. A368052, A368481, A368483, A006722, A369113, A212979.

(Maxima) N : 6$ Len : 12$     /* Somos-N, N >= 2, Len = length of the calculated lists */
NofRT : floor (N / 2)$  /* number of terms in a Somos-N recurrence */
S : makelist (0, Len)$
G : makelist (0, Len)$ DegG : makelist (0, Len)$   /* G, the numerator of s() */
for i: 1 thru N do ( S[i] : s[i - 1], G[i] : 1, DegG[i] : 0 )$
for i: N + 1 thru Len do (
   SS : 0,
   for j : 1 thru NofRT do (
      SS : SS + S[i - j] * S[i - N + j]
   ),
   S[i] : factor (SS / S[i - N]), G[i] : num (S[i]),
   /* for N > 3 G is a homogenous polynomial, take the first monomial to determine the degree */
   Mon : G[i], if N > 3 then ( Mon : args (Mon)[1] ),
   DegG[i] : 0, for j : 0 thru N - 1 do ( DegG[i] : DegG[i] + hipow (Mon, s[j]) )
)$ DegG;

a(n) = 1 + e(n-5) + e(n-4) + e(n-3) + e(n-2) + e(n-1) + e(n), where e(n) = A369113(n) is the exponent of one of the initial values in the denominator of s(n).
The growth rate is quadratic, a(n) = (3/20) * n^2 + O(n).
For n > 5 a(n) = (A212979(n-3) - 4) / 3. - Helmut Ruhland, Jan 31 2024

cp's OEIS Frontend

A369113 Tropical version of Somos-6 sequence A006722.

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A141605 Triangle read by rows: T(n, k) = A006722(n)/(A006722(k)*A006722(n-k)).

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A276402 A sequence related to the Somos-6 sequence A006722.

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A038754 a(2n) = 3^n, a(2n+1) = 2*3^n.

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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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