A006720 -id:A006720 - cp's OEIS frontend

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.

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

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

A006769 Elliptic divisibility sequence associated with elliptic curve "37a1": y^2 + y = x^3 - x and multiples of the point (0,0).

Original entry on oeis.org

0, 1, 1, -1, 1, 2, -1, -3, -5, 7, -4, -23, 29, 59, 129, -314, -65, 1529, -3689, -8209, -16264, 83313, 113689, -620297, 2382785, 7869898, 7001471, -126742987, -398035821, 1687054711, -7911171596, -47301104551, 43244638645

Offset: 0

Michael Somos, Jul 16 1999

This sequence has a recursion same as the Somos-4 sequence recursion.
a(n+1) is the Hankel transform of A178072. - Paul Barry, May 19 2010
The recurrence formulas in [Kimberling, p. 16] are missing square and cube exponents. - Michael Somos, Jul 07 2014
This is a strong elliptic divisibility sequence t_n as given in [Kimberling, p. 16] where x = 1, y = -1, z = 1.
From Helmut Ruhland, Nov 28 2023: (Start)
This sequence and its two subsequences with even/odd indices satisfy the Somos-4 recursion.
The even subsequence is A051138, here called r[ ]. The odd subsequence is the classical Somos-4 A006720, here called s[ ].
These two subsequences interleaved as follows, recover the original sequence which is now: r[0], s[2], r[1], -s[3], r[2], s[4], r[3], -s[5], ...,  all Somos-4 s[ ] with odd index with a minus sign. (End)

G. Everest, A. van der Poorten, I. Shparlinski and T. Ward, Recurrence Sequences, Amer. Math. Soc., 2003; pp. 11 and 164.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Seiichi Manyama, Table of n, a(n) for n = 0..300 (first 101 terms from T. D. Noe)
Paul Barry, Riordan Pseudo-Involutions, Continued Fractions and Somos 4 Sequences, arXiv:1807.05794 [math.CO], 2018.
Paul Barry, Generalized Catalan recurrences, Riordan arrays, elliptic curves, and orthogonal polynomials, arXiv:1910.00875 [math.CO], 2019.
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.
Graham Everest, Elliptic Divisibility Sequences.
R. W. Gosper and Richard C. Schroeppel, Somos Sequence Near-Addition Formulas and Modular Theta Functions, arXiv:math/0703470 [math.NT], 2007.
Clark Kimberling, Strong divisibility sequences and some conjectures, Fib. Quart., 17 (1979), 13-17.
LMFDB, Elliptic Curve 37.a1 (Cremona label 37a1)
Helmut Ruhland, Somos-4 and a quartic Surface in RP^3, arXiv:2312.02085 [math.AG], 2023.
Michael Somos, Number Walls in Combinatorics
Index entries for two-way infinite sequences

Cf. A006720, A028941, A050512, A051138, A178072, A278314.

a006769 n = a050512_list !! n
a006769_list = 0 : 1 : 1 : (-1) : 1 : zipWith div (zipWith (+) (zipWith (*)
   (drop 4 a006769_list) (drop 2 a006769_list))
     (map (^ 2) (drop 3 a006769_list))) (tail a006769_list)
-- Reinhard Zumkeller, Nov 02 2011

a[n_] := If[n < 0, -a[-n], If[n == 0, 0, ClearAll[an]; an[] = 1; an[3] = -1; For[k = 5, k <= n, k++, an[k] = (an[k-1]*an[k-3] + an[k-2]^2)/an[k-4]]; an[n]]]; Table[a[n], {n, 0, 32}] (* _Jean-François Alcover, Dec 14 2011, after first Pari program *)
Join[{0},RecurrenceTable[{a[1]==a[2]==1,a[3]==-1,a[4]==1,a[n]==(a[n-1] a[n-3]+ a[n-2]^2)/a[n-4]},a,{n,40}]] (* Harvey P. Dale, May 04 2018 *)
a[ n_] := Which[n<0, -a[-n], n<5, {0, 1, 1, -1, 1}[[1+n]], True, (a[n-1]*a[n-3] + a[n-2]^2)/a[n-4]]; (* Michael Somos, Aug 20 2024 *)

{a(n) = my(an); if( n<0, -a(-n), if( n==0, 0, an = vector( max(3, n), i, 1); an[3] = -1; for( k=5, n, an[k] = (an[k-1] * an[k-3] + an[k-2]^2) / an[k-4]); an[n]))};

{a(n) = my(an); if( n<0, -a(-n), if( n==0, 0, an = Vec((-1 - 2*x + sqrt(1 + 4*x - 4*x^3 + O(x^n))) / (2 * x^2)); matdet( matrix((n-1)\2, (n-1)\2, i, j, if(i + j - 1 - n%2<0, 0, an[i + j -n%2])))))};

{a(n) = my(E, z); E = ellinit([0, 0, -1, -1, 0]); z = ellpointtoz(E, [0, 0]); round( ellsigma(E, n*z) / ellsigma(E, z)^(n^2))}; /* Michael Somos, Oct 22 2004 */

{a(n) = sign(n) * subst( elldivpol( ellinit([0, 0, -1, -1, 0]), abs(n)), x, 0)}; /* Michael Somos, Dec 16 2014 */

a(n) = (a(n-1) * a(n-3) + a(n-2)^2) / a(n-4) for all n != 4.
a(n) = (-a(n-1) * a(n-4) - a(n-2) * a(n-3)) / a(n-5) for all n != 5.
a(-n) = -a(n) for all n.
a(2*n + 1) = a(n+2) * a(n)^3 - a(n-1) * a(n+1)^3, a(2*n) = a(n+2) * a(n) * a(n-1)^2 - a(n) * a(n-2) * a(n+1)^2 for all n.
A006720(n) = (-1)^n * a(2*n - 3), A028941(n) = a(n)^2 for all n.
a(2*n) = A051138(n). - Michael Somos, Feb 10 2015
a(2*n+1) = a(n-1)*a(n)^2*a(n+3) - a(n-2)*a(n+1)^2*a(n+2) for all n. - Michael Somos, Aug 20 2024

A060007 Decimal expansion of the positive real root of x^4 - x - 1.

Original entry on oeis.org

1, 2, 2, 0, 7, 4, 4, 0, 8, 4, 6, 0, 5, 7, 5, 9, 4, 7, 5, 3, 6, 1, 6, 8, 5, 3, 4, 9, 1, 0, 8, 8, 3, 1, 9, 1, 4, 4, 3, 2, 4, 8, 9, 0, 8, 6, 2, 4, 8, 6, 3, 5, 2, 1, 4, 2, 8, 8, 2, 4, 4, 4, 5, 3, 0, 4, 9, 7, 1, 0, 0, 0, 8, 5, 2, 2, 5, 9, 1, 3, 5, 0, 2, 5, 3, 0, 9, 5, 5, 2, 1, 8, 6, 9, 9, 6, 2, 8, 6, 2, 5, 7, 4, 0, 1

Offset: 1

Fabian Rothelius, Mar 14 2001

Original name: Decimal expansion of v_4, where v_n is the smallest, positive, real solution to the equation (v_n)^n = v_n + 1.
v_2 = A001622 - 1. [Corrected by M. F. Hasler, Jul 12 2025]
v_3 = A060006, a.k.a. plastic constant, real root of x^3 - x - 1. - M. F. Hasler, Jul 12 2025
A Perron number of the 4th degree polynomial (see Boys and Wu). - R. J. Mathar, Mar 19 2011
This number is not ruler-and-compass constructible because x^4-x-1 and its resolvent x^3+4x+1 are irreducible over the rationals. - Jean-François Alcover, Aug 31 2015
The other (negative) real root -0.724491959... is -A356032. The first of the pair of complex conjugate roots is obtained by negating in the formula for v_4 below sqrt(2*u) and sqrt(u), giving -0.2481260628... - 1.0339820609...*i. - Wolfdieter Lang, Aug 27 2022
The sequence a(n) = v_4^((n^2-n)/2) satisfies the Somos-4 recursion a(n+2)*a(n-2) = a(n+1)*a(n-1) + a(n)^2 for all n in Z. - Michael Somos, Mar 24 2023

			v_4 = 1.220744084605759475361685349...

Vincenzo Librandi, Table of n, a(n) for n = 1..1000
David W. Boys, The maximal modulus of an algebraic integer, Math. Comp. 45 (1985) 243-249, table page S18.
F. Rothelius, Formulae
Qiang Wu, The smallest Perron numbers, Math. Comp. 79 (2010) 2387-2394
Index entries for algebraic numbers, degree 4

Cf. A001622 (golden ratio, root of x^2 - x - 1), A060006 (plastic number, root of x^3 - x - 1), A202540 (log thereof), A160155 (root of x^5 - x - 1), A356032 (root of x^4 + x - 1), A006720, A298813.

r:=(108+12*sqrt(849))^(1/3): (sqrt(12/sqrt(-8/r+r/6)+48/r-r) + sqrt(-48/r+r))/(2*sqrt(6)): evalf(%,105); # Vaclav Kotesovec, Oct 12 2013

RealDigits[x/.FindRoot[x^4==x+1,{x,1},WorkingPrecision->120]][[1]] (* Harvey P. Dale, Jul 11 2012 *)
Root[ #^4 - # - 1&, 2] // RealDigits[#, 10, 105]& // First (* Jean-François Alcover, Mar 04 2013 *)

default(realprecision, 110); digits(floor(solve(x=1, 2, x^4 - x - 1)*10^105)) /* Michael Somos, Mar 22 2023 */

Equals (1 + (1 + (1 + (1 + (1 + ...)^(1/4))^(1/4))^(1/4))^(1/4))^(1/4). - Ilya Gutkovskiy, Dec 15 2017
v_4 = (sqrt(2)*u + sqrt(sqrt(2*u) - 2*u^2))/(2*sqrt(u)), with u = (Ap^(1/3) + ep*Am^(1/3))/3, where Ap = (3/16)*(9 + sqrt(3*283)), Am = (3/16)*(9 - sqrt(3*283)), ep = (-1 + sqrt(3)*i)/2 and i = sqrt(-1).
For the trigonometric equivalent u = (2/3)*sqrt(3)*sinh((1/3)*arcsinh((3/16)* sqrt(3))). - Wolfdieter Lang, Aug 27 2022
Equals 1 + Sum_{n >= 1} (1/4)^n*(Product_{j=1..n-1} 1 + n - 4*j)/n!. - Antonio Graciá Llorente, Dec 13 2024
Equals exp(A202540) = sqrt(A298813). - Hugo Pfoertner, Dec 14 2024

More terms from Benoit Cloitre, Jan 11 2003

Simplified definition from M. F. Hasler, Jul 12 2025

A072879 a(n) = 5a(n-1)a(n-2)a(n-3)a(n-4) - a(n-5) with a(1) = a(2) = a(3) = a(4) = a(5) = 1.

Original entry on oeis.org

1, 1, 1, 1, 1, 4, 19, 379, 144019, 20741616379, 107553662508585672001, 608831069421618273050865038881215685876, 978035016076705458999330010986670207956236476587064788804921180339451725001

Offset: 1

Benoit Cloitre, Jul 28 2002

Solutions of the Hurwitz equation in five variables.

Seiichi Manyama, Table of n, a(n) for n = 1..16
Arthur Baragar, Integral solutions of the Markoff-Hurwitz equations, J. Number Theory 49 (1994), 27-44.
Andrew N. W. Hone, Diophantine non-integrability of a third order recurrence with the Laurent property, arXiv:math/0601324 [math.NT], 2006.
Andrew N. W. Hone, Diophantine non-integrability of a third order recurrence with the Laurent property, J. Phys. A: Math. Gen. 39 (2006), L171-L177.
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.

Cf. A006720, A064098, A072878, A072880.

nxt[{a_,b_,c_,d_,e_}]:={b,c,d,e,(5b c d e)-a}; NestList[nxt,{1,1,1,1,1},20][[All,1]] (* Harvey P. Dale, Nov 07 2016 *)

a(1) = a(2) = a(3) = a(4) = a(5) = 1; a(n) = (a(n-1)^2+a(n-2)^2+a(n-3)^2+a(n-4)^2)/a(n-5) for n >= 6.

From the recurrence a(n) = 5*a(n-1)*a(n-2)*a(n-3)*a(n-4) - a(n-5), any five successive terms satisfy the five-variable Hurwitz equation a(n)^2+a(n-1)^2+a(n-2)^2+a(n-3)^2+a(n-4)^2 = 5*a(n)*a(n-1)*a(n-2)*a(n-3)*a(n-4). As n tends to infinity, the limit of log(log(a(n)))/n is log x = 0.6562559790..., where x=1.927561975... is the largest real root of the quartic x^4-x^3-x^2-x-1=0. - Andrew Hone, Nov 16 2005

Entry revised Nov 19 2005, based on comments from Andrew Hone

Name clarified by Petros Hadjicostas, May 11 2019

A072880 A recurrence of order 6: a(1)=a(2)=a(3)=a(4)=a(5)=a(6)=1; a(n) = (a(n-1)^2 + a(n-2)^2 + a(n-3)^2 + a(n-4)^2 + a(n-5)^2)/a(n-6).

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 5, 29, 869, 756029, 571580604869, 326704387862983487112029, 21347151409785350408171299054974277225256721769, 15713823217665540462976624783900822313284439536736221766688609460305249837839107387688348185

Offset: 1

Benoit Cloitre, Jul 28 2002

Any six successive terms satisfy the Markoff-Hurwitz equation a^2 + b^2 + c^2 + d^2 + e^2 + f^2 = 6*a*b*c*d*e*f. - Bruno Langlois, Aug 09 2016

Seiichi Manyama, Table of n, a(n) for n = 1..17
Andrew N. W. Hone, Diophantine non-integrability of a third order recurrence with the Laurent property, arXiv:math/0601324 [math.NT], 2006.
Andrew N. W. Hone, Diophantine non-integrability of a third order recurrence with the Laurent property, J. Phys. A: Math. Gen. 39 (2006), L171-L177.
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.

Cf. A006720, A064098, A072878, A072879.

RecurrenceTable[{a[n] == (a[n - 1]^2 + a[n - 2]^2 + a[n - 3]^2 + a[n -  4]^2 + a[n - 5]^2)/a[n - 6], a[1] == a[2] == a[3] == a[4] == a[5] == a[6] == 1}, a, {n, 1, 14}] (* Michael De Vlieger, Aug 11 2016 *)
nxt[{a_, b_, c_, d_, e_, f_}] := {b, c, d, e, f,(b^2+c^2+d^2+e^2+f^2)/a}; NestList[ nxt, Table[1,6],20][[All,1]] (* Harvey P. Dale, Mar 18 2018 *)

a(n) = 6*a(n-1)*a(n-2)*a(n-3)*a(n-4)*a(n-5) - a(n-6). - Bruno Langlois, Aug 09 2016

A018896 a(n) = ( a(n-1)*a(n-7) + a(n-4)^2 ) / a(n-8); a(0) = ... = a(7) = 1.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 2, 3, 4, 5, 9, 18, 34, 93, 180, 348, 724, 3033, 9666, 24986, 83761, 261033, 1023728, 3923791, 26128126, 105734485, 381740209, 1895904805, 14058722881, 97964968321, 517832518189, 4364261070929, 25225712161101, 181840424632390

Offset: 0

Michael Somos

From Vladimir Shevelev, Apr 04 2016: (Start)
For k >= 0, an infinite sequence {a(k,n)} of Somos's sequences (n>=0) is:
a(k,0) = a(k,1)= ... = a(k,2*k+1) = 1;
and then for n >= 2*k+2,
a(k,n) = (a(k,n-1)*a(k,n-2*k-1) + a(k,n-k-1)^2)/a(k,n-2*k-2).
In particular, {a(0,n)}=A006125, {a(1,n)}=A006720, {a(2,n)}=A102276, {a(3,n)}=A018896.
One can prove that the sequence {a(k,n)} has the first 4k+2 simple differences: 2k+1 zeros, after that k+1 1's and after that k consecutive squares, beginning with 2^2.
Further we have nontrivial differences. The first of them for k=0,1,2,... are 6, 16, 33, 59, 96, 146, 211, 293, 394, 516, ... that is, {k^3/3 + 5*k^2/2 + 43*k/6 + 6}.
(End)

T. D. Noe, Table of n, a(n) for n = 0..100
Mohamed Bensaid, Sato tau functions and construction of Somos sequence, arXiv:2409.05911 [math.NT], 2024. See p. 7.
David Gale, Mathematical Entertainments, Mathematical Intelligencer, volume 18, number 3, Summer 1996, page 25.
Eric Weisstein's World of Mathematics, Somos Sequence
Index entries for two-way infinite sequences

Cf. A006125, A006720, A102276.

a018896 n = a018896_list !! n
a018896_list = replicate 8 1 ++ f 8 where
   f x = ((a018896 (x - 1) * a018896 (x - 7) + a018896 (x - 4) ^ 2)
         `div` a018896 (x - 8)) : f (x + 1)
-- Reinhard Zumkeller, Oct 01 2012

[n le 8 select 1 else (Self(n-1)*Self(n-7)+Self(n-4)^2 ) / Self(n-8): n in [1..40]]; // Vincenzo Librandi, Dec 08 2016

f:= proc(n) option remember;
  if n <= 7 then 1 else
  (procname(n-1)*procname(n-7)+procname(n-4)^2)/procname(n-8)
  fi
end proc:
seq(f(n),n=0..50); # Robert Israel, Apr 04 2016

RecurrenceTable[{a[1]==a[2]==a[3]==a[4]==a[5]==a[6]==a[7]==a[8]==1, a[n]==(a[n-1]a[n-7]+ a[n-4]^2)/a[n-8]},a[n],{n,50}] (* Harvey P. Dale, May 02 2011 *)
k = 3; Set[#, 1] & /@ Map[a[k, #] &, Range[0, 2 k + 1]]; a[k_, n_] /; n >= 2 k + 2 := (a[k, n - 1] a[k, n - 2 k - 1] + a[k, n - k - 1]^2)/ a[k, n - 2 k - 2]; Table[a[k, n], {n, 0, 35}] (* Michael De Vlieger, Apr 04 2016 *)

More terms from Harvey P. Dale, May 02 2011

A051138 Divisibility sequence associated with elliptic curve y^2 + y = x^3 - x and point (1, 0).

Original entry on oeis.org

0, 1, 1, -1, -5, -4, 29, 129, -65, -3689, -16264, 113689, 2382785, 7001471, -398035821, -7911171596, 43244638645, 6480598259201, 124106986093951, -5987117709349201, -541051130050800400, -4830209396684261199

Offset: 0

Michael Somos, Oct 12 1999

This is a strong divisibility sequence; that is, if n divides m, then a(n) divides a(m) and moreover for all positive integer n,m a(gcd(n, m)) = gcd(a(n), a(m)).
This is a strong elliptic divisibility sequence t_n as given in [Kimberling, p. 16] where x = 1, y = -1, z = -5. - Michael Somos, Jul 07 2014
The elliptic curve y^2 + y = x^3 - x has LMFDB label 37.a1 (Cremona label 37a1). - Michael Somos, Feb 07 2024

			G.f. = x + x^2 - x^3 - 5*x^4 - 4*x^5 + 29*x^6 + 129*x^7 - 65*x^8 + ...

T. D. Noe, Table of n, a(n) for n = 0..100
Clark Kimberling, Strong divisibility sequences and some conjectures, Fib. Quart., 17 (1979), 13-17.
R. W. Gosper and Richard C. Schroeppel, Somos Sequence Near-Addition Formulas and Modular Theta Functions, arXiv:math/0703470 [math.NT], 2007.
LMFDB, Elliptic Curve 37.a1
Helmut Ruhland, Somos-4 and a quartic Surface in RP^3, arXiv:2312.02085 [math.AG], 2023.
Index to divisibility sequences

Cf. A006769, A006720, A028937, A028939.

a[n_ /; n < 0] := -a[-n]; a[0] = 0; A006769[n_] := (ClearAll[an]; an[] = 1; an[3] = -1; For[k = 5, k <= n, k++, an[k] = (an[k-1]*an[k-3] + an[k-2]^2)/an[k-4]]; an[n]); a[n] := A006769[2n]; Table[a[n], {n, 0, 21}] (* Jean-François Alcover, Apr 11 2012, after 2nd formula *)

an = vector(200); an = concat([ 1, 1, -1, -5 ], an); for( n=5, length(an), an[ n ]=(an[ n-1 ] * an[ n-3 ] + an[ n-2 ]^2) / an[ n-4 ]); a(n) = an[ n ]

{a(n) = my(v = [1, 1, -1, -5]); if( n<0, -a(-n), if( n==0, 0, if( n<5, v[n], v = concat( v, vector(n - 4)); for( k=5, n, v[k] = (v[k-1] * v[k-3] + v[k-2]^2) / v[k-4]); v[n])))}; /* Michael Somos, Feb 12 2012 */

{a(n) = subst(elldivpol(ellinit([0, 0, 1, -1, 0]), n, 'x), 'x, 1)}; /* Michael Somos, Apr 21 2025 */

a(n) = (a(n-1) * a(n-3) + a(n-2)^2) / a(n-4).
a(n) = (-a(n-1) * a(n-4) + 5 * a(n-2) * a(n-3)) / a(n-5).
a(2*n + 1) = a(n+2) * a(n)^3 - a(n-1) * a(n+1)^3.
a(2*n) = a(n+2) * a(n) * a(n-1)^2 - a(n) * a(n-2) * a(n+1)^2.
a(-n) = -a(n). a(n) = A006769(2*n). a(n)^2 = A028937(n). |a(n)|^3 = A028939(n) for all n in Z.
0 = a(n)*a(n+4) - a(n+1)*a(n+3) - a(n+2)*a(n+2) for all n in Z. - Michael Somos, Jul 07 2014
0 = a(n)*a(n+5) + a(n+1)*a(n+4) - 5*a(n+2)*a(n+3) for all n in Z. - Michael Somos, Jul 07 2014

A271831 Somos's sequence {a(6,n)} defined in comment in A018896: a(0)=a(1)= ... = a(13) = 1; for n>=14, a(n) = (a(n-1)*a(n-13) + a(n-7)^2)/a(n-14).

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 3, 4, 5, 6, 7, 8, 12, 21, 37, 62, 98, 147, 358, 609, 959, 1541, 2618, 4655, 8407, 28631, 81011, 186528, 376741, 706041, 1280174, 3598503, 8411236, 24021605, 74880071, 219318499, 580374907, 1400227135, 6308924342

Offset: 0

Vladimir Shevelev and Peter J. C. Moses, Apr 15 2016

nonn

Seiichi Manyama, Table of n, a(n) for n = 0..416
Eric Weisstein's World of Mathematics, Somos Sequence

Cf. A018896, A006125, A006720, A102276, A271341, A271835, A271837, A271838, A271839.

[n le 14 select 1 else (Self(n-1)*Self(n-13) + Self(n-7)^2)/Self(n-14): n in [1..50]]; // G. C. Greubel, Feb 21 2018

a[n_ /; 0 <= n <= 14] = 1; a[n_]:= a[n] = (a[n-1]*a[n-13] + a[n-7]^2)/a[n -14]; Table[a[n], {n,0,50}] (* G. C. Greubel, Feb 21 2018 *)

{a(n) = if(n<= 14, 1, (a(n-1)*a(n-13) + a(n-7)^2)/a(n-14))};
for(n=1,50, print1(a(n), ", ")) \\ G. C. Greubel, Feb 21 2018

cp's OEIS Frontend

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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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.

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A006769 Elliptic divisibility sequence associated with elliptic curve "37a1": y^2 + y = x^3 - x and multiples of the point (0,0).

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A060007 Decimal expansion of the positive real root of x^4 - x - 1.

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A072879 a(n) = 5*a(n-1)*a(n-2)*a(n-3)*a(n-4) - a(n-5) with a(1) = a(2) = a(3) = a(4) = a(5) = 1.

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A072879 a(n) = 5a(n-1)a(n-2)a(n-3)a(n-4) - a(n-5) with a(1) = a(2) = a(3) = a(4) = a(5) = 1.