A006769 -id:A006769 - cp's OEIS frontend

A278314 a(n) = -c(n-1) * c(n-2) * c(n+3) where c(n) = A006769(n).

0, 0, 1, -3, -5, -14, -8, 69, -435, 2065, 3612, 28888, -43355, -2616119, 28076979, -332513754, 331948240, 8280062505, 641260644409, 18784454671297, -318128427505160, 10663732503571536, -66316334575107447, -8938035295591025771, -588310630753491921045

Offset: 1

Michael Somos, Nov 17 2016

sign

a(n) = A028942(n) up to sign.

y coordinate of n*P = -A028942(n) / A028943(n) = a(n) / A006769(n)^3 where P is generator for rational points on curve y^2 + y = x^3 - x.

			G.f. = x^3 - 3*x^4 - 5*x^5 - 14*x^6 - 8*x^7 + 69*x^8 - 435*x^9 + ...

Seiichi Manyama, Table of n, a(n) for n = 1..173

Cf. A006769, A028942, A028943.

{a(n) = my(m, an); if( n>0, m = n; an = vector( max(12, m), i, if( i<13, [0, 0, 1, -3, -5, -14, -8, 69, -435, 2065, 3612, 28888][i], 0)), m = 1-n; an = vector( max(12, m), i, if( i<13, [1, 1, 1, 0, -2, 3, -15, -35, -56, -92, 2001, -8555][i], 0))); for( k=13, m, an[k] = (an[k-1] * an[k-7] + 3 *  an[k-2] * an[k-6] - 3 * an[k-3] * an[k-5] + 6 * an[k-4]^2) / an[k-8]); an[m]};

0 = a(n)*a(n+8) - a(n+1)*a(n+7) - 3*a(n+2)*a(n+6) + 3*a(n+3)*a(n+5) - 6*a(n+4)^2 for all n in Z.

0 = a(n+1)*a(n+2)*a(n+6) - 2*a(n+1)*a(n+3)*a(n+5) + 3*a(n+1)*a(n+4)^2 + 3*a(n+2)^2*a(n+5) + a(n+2)*a(n+3)*a(n+4) - a(n+3)^3 for all n in Z.

A178072 Sequence with Hankel transform equal to the Somos-4 sequence A006769(n+2).

Original entry on oeis.org

1, 0, -1, -1, -1, -1, 1, 8, 23, 45, 55, -14, -317, -1095, -2459, -3574, -681, 16124, 64605, 159483, 260869, 134374, -906919, -4228769, -11317061, -20327731, -15742753, 52640154, 293447719, 847451759, 1648865921, 1636313816, -2986606297, -21074495982

Offset: 0

Paul Barry, May 19 2010

			G.f. = 1 - x^2 - x^3 - x^4 - x^5 + x^6 + 8*x^7 + 23*x^8 + 45*x^9 + 55*x^10 + ...

G. C. Greubel, Table of n, a(n) for n = 0..1000
Paul Barry, Integer sequences from elliptic curves, arXiv:2306.05025 [math.NT], 2023.

Cf. A006769.

m:=40; R:=PowerSeriesRing(Rationals(), m); Coefficients(R!(2/(1+2*x+x^2+Sqrt(1-4*x+6*x^2+x^4)))); // G. C. Greubel, Sep 22 2018

CoefficientList[Series[2/(1+2*x+x^2+Sqrt[1-4*x+6*x^2+x^4]), {x, 0, 40}], x] (* G. C. Greubel, Sep 22 2018 *)

{a(n) = if( n<0, 0, polcoeff( 2 / (1 + 2*x + x^2 + sqrt(1 - 4*x + 6*x^2 + x^4 + x*O(x^n))), n))}; /* Michael Somos, Jun 13 2010 */

my(x='x+O('x^50)); Vec(2/(1+2*x+x^2+(1-4*x+6*x^2+x^4)^(1/2))) \\ Altug Alkan, Sep 23 2018

a(n) = Sum_{k=0..floor(n/2)} (C(n-k,k)/(n-2*k+1))*Sum_{i=0..min(k,n-2*k)} C(k,i)*C(n-k-i-1,n-2*k-i)*2^(n-2*k-i)*(-1)^(k-i).
From Michael Somos, Jun 13 2010: (Start)
G.f. A(x) satisfies 0 = f(x, A(x)) where f(x, y) = (2*x + x^3) * y^2 - (1 + x)^2 * y + 1.
G.f.: 2 / (1 + 2*x + x^2 + sqrt(1 - 4*x + 6*x^2 + x^4)). (End)
Conjecture: 2*n*(n+1)*a(n) +2*n*(n+1)*a(n-1) -3*(3*n-1)*(3*n-4)*a(n-2) +(61*n^2-191*n+36)*a(n-3) +6*(-2*n^2+2*n-1)*a(n-4) +2*(5*n-1)*(4*n-15)*a(n-5) +n*(n-5)*a(n-6) +(5*n-1)*(n-6)*a(n-7)=0. - R. J. Mathar, Jun 14 2016

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

A025262 a(n) = a(1)a(n-1) + a(2)a(n-2) + ... + a(n-1)*a(1) for n >= 4.

Original entry on oeis.org

1, 1, 1, 3, 8, 23, 68, 207, 644, 2040, 6558, 21343, 70186, 232864, 778550, 2620459, 8872074, 30195288, 103246502, 354508628, 1221846856, 4225644866, 14659644348, 51002664023, 177909901566, 622093882290, 2180123564130, 7656055966092

Offset: 1

Clark Kimberling

nonn

a(n) is the number of ascent sequences (A022493) of length n-1 such that the nonzero entries are weakly increasing and no two consecutive entries are both 0. For example a(4) = 3 counts 010, 011, 012 and a(5) = 8 counts 0101, 0102, 0110, 0111, 0112, 0120, 0122, 0123. - David Callan, Nov 25 2021

The o.g.f. y (= x + x^2 + x^3 + ...) of this sequence satisfies y^2 - y = x^3 - x. If y is replaced by -y, then it is the elliptic curve y^2 + y = x^3 - x with LMFDB label 37.a1 (Cremona label 37a1) associated to the Somos-4 sequence via elliptic divisibility sequence A006769. - Michael Somos, Apr 18 2023

			G.f. = x + x^2 + x^3 + 3*x^4 + 8*x^5 + 23*x^6 + 68*x^7 + 207*x^8 + 644*x^9 + ...

Seiichi Manyama, Table of n, a(n) for n = 1..1766
Paul Barry, On the Hurwitz Transform of Sequences, Journal of Integer Sequences, Vol. 15 (2012), #12.8.7.
Paul Barry, Integer sequences from elliptic curves, arXiv:2306.05025 [math.NT], 2023.
David Callan, On Ascent, Repetition and Descent Sequences, arXiv:1911.02209 [math.CO], 2019.
Michael Somos, Number Walls in Combinatorics.
Fumitaka Yura, Hankel Determinant Solution for Elliptic Sequence, arXiv:1411.6972 [nlin.SI], 2014; see p. 7.

Cf. A000108, A176703, A056010, A025268, A006769.

nmax = 30; aa = ConstantArray[0, nmax]; aa[[1]] = 1; aa[[2]] = 1; aa[[3]] = 1; Do[aa[[n]] = Sum[aa[[k]] * aa[[n - k]], {k, 1, n - 1}], {n, 4, nmax}]; aa (* Vaclav Kotesovec, Jan 25 2015 *)
Nest[Append[#, #.Reverse[#]] &, {1, 1, 1}, 25] (* Jan Mangaldan, Jul 07 2020 *)

{a(n) = polcoeff( (1 - sqrt(1 - 4*x + 4*x^3 + x * O(x^n))) / 2, n)}; /* Michael Somos, Aug 04 2000 */

G.f.: (1 - sqrt(1 - 4*x + 4*x^3)) / 2. Satisfies A(x) - A(x)^2 = x - x^3. - Michael Somos, Aug 04 2000
Given an integer t >= 1 and initial values u = [a_0, a_1, ..., a_{t-1}], we may define an infinite sequence Phi(u) by setting a_n = a_{n-1} + a_0*a_{n-1} + a_1*a_{n-2} + ... + a_{n-2}*a_1 for n >= t. For example Phi([1]) is the Catalan numbers A000108. The present sequence is Phi([1,1,1]). - Gary W. Adamson, Oct 27 2008
Row sums of A176703 if offset 0. - Michael Somos, Jan 09 2012
a(n+2) = A056010(n) if n >= 0.
a(n) = Sum_{m=0..floor((n-1)/2)} C(n-2*m-1)*binomial(n-2*m,m)*(-1)^m, where C = A000108 are the Catalan numbers. - Vladimir Kruchinin, Jan 26 2013
0 = a(n)*(+16*a(n+1) - 64*a(n+3) + 22*a(n+4)) + a(n+1)*(+32*a(n+2) - 14*a(n+3)) + a(n+2)*(+16*a(n+3) - 10*a(n+4)) + a(n+3)*(+2*a(n+3) + a(n+4)) if n>0. - Michael Somos, Jan 18 2015
Recurrence: n*a(n) = 2*(2*n-3)*a(n-1) - 2*(2*n-9)*a(n-3). - Vaclav Kotesovec, Jan 25 2015
a(n) ~ sqrt(3 - 8*r) * (4 - 4*r^2)^n / (4*sqrt(Pi)*n^(3/2)), where r = 2*sin(arccos(-3^(3/2)/8)/3 - Pi/6)/sqrt(3). - Vaclav Kotesovec, Jun 05 2022

A028940 a(n) = numerator of the X-coordinate of n*P where P is the generator [0,0] for rational points on the curve y^2 + y = x^3 - x.

Original entry on oeis.org

0, 1, -1, 2, 1, 6, -5, 21, -20, 161, 116, 1357, -3741, 18526, 8385, 480106, -239785, 12551561, -59997896, 683916417, 1849037896, 51678803961, -270896443865, 4881674119706, -16683000076735, 997454379905326

Offset: 1

N. J. A. Sloane

We can take P = P[1] = [x_1, y_1] = [0,0]. Then P[n] = P[1]+P[n-1] = [x_n, y_n] for n >= 2. Sequence gives numerators of the x_n. - N. J. A. Sloane, Jan 27 2022

			4P = P[4] = [2, -3].
P[1] to P[16] are [0, 0], [1, 0], [-1, -1], [2, -3], [1/4, -5/8], [6, 14], [-5/9, 8/27], [21/25, -69/125], [-20/49, -435/343], [161/16, -2065/64], [116/529, -3612/12167], [1357/841, 28888/24389], [-3741/3481, -43355/205379], [18526/16641, -2616119/2146689], [8385/98596, -28076979/30959144], [480106/4225, 332513754/274625]. - _N. J. A. Sloane_, Jan 27 2022

A. W. Knapp, Elliptic Curves, Princeton 1992, p. 77.

Seiichi Manyama, Table of n, a(n) for n = 1..212
B. Mazur, Arithmetic on curves, Bull. Amer. Math. Soc. 14 (1986), 207-259; see p. 225.

Cf. A006769, A028941, A028942, A028943, A350622-A350625.

\\ from N. J. A. Sloane, Jan 27 2022. To get the first 40 points P[n].
\\ define curve E
E = ellinit([0,0,1,-1,0]) \\ y^2+y = x^3-x
P = vector(100)
P[1] = [0,0]
for(n=2, 40, P[n] = elladd(E, P[1], P[n-1]))
P

P = (0, 0), 2P = (1, 0); if kP = (a, b) then (k+1)P = (a' = (b^2 - a^3)/a^2, b' = -1 - b*a'/a).

a(-n) = a(n) = - A006769(n-1) * A006769(n+1) for all n in Z. - Michael Somos, Jul 28 2016

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

A028941 Denominator of X-coordinate of n*P where P is the generator [0,0] for rational points on curve y^2+y = x^3-x.

Original entry on oeis.org

1, 1, 1, 1, 4, 1, 9, 25, 49, 16, 529, 841, 3481, 16641, 98596, 4225, 2337841, 13608721, 67387681, 264517696, 6941055969, 12925188721, 384768368209, 5677664356225, 61935294530404, 49020596163841, 16063784753682169

Offset: 1

N. J. A. Sloane

We can take P = P[1] = [x_1, y_1] = [0,0]. Then P[n] = P[1]+P[n-1] = [x_n, y_n] for n >= 2. Sequence gives denominators of the x_n. - N. J. A. Sloane, Jan 27 2022

Squares of terms in A006769 (or A006720).

G. Everest, A. J. van der Poorten, I. Shparlinski and T. Ward, Recurrence Sequences: Examples and Applications, AMS Monographs, 2003
A. W. Knapp, Elliptic Curves, Princeton 1992, p. 77.

Seiichi Manyama, Table of n, a(n) for n = 1..212
B. Mazur, Arithmetic on curves, Bull. Amer. Math. Soc. 14 (1986), 207-259; see p. 225.

Cf. A028940, A028942, A028943.

nxt[{a_,b_,c_,d_,e_,f_}]:={b,c,d,e,f,((c*2e)-f*b+2d^2)/a}; NestList[nxt,{1,1,1,1,4,1},30][[;;,1]] (* Harvey P. Dale, Dec 26 2024 *)

PARI
```
- see A028940.
```

This sequence satisfies the quadratic recurrence relation a(n)a(n-6)=-a(n-1)a(n-5)+2a(n-2)a(n-4)+2a(n-3)^2 which is a generalized Somos-6 relation. - Graham Everest (g.everest(AT)uea.ac.uk), Dec 16 2002

P=(0, 0), 2P=(1, 0), if kP=(a, b) then (k+1)P=(a'=(b^2-a^3)/a^2, b'=-1-b*a'/a).

A056010 Number of words of length n in a simple grammar.

Original entry on oeis.org

1, 1, 3, 8, 23, 68, 207, 644, 2040, 6558, 21343, 70186, 232864, 778550, 2620459, 8872074, 30195288, 103246502, 354508628, 1221846856, 4225644866, 14659644348, 51002664023, 177909901566, 622093882290, 2180123564130, 7656055966092

Offset: 0

Michael Somos, Aug 01 2000

nonn

The grammar defines a language L consisting of words from the alphabet S = {e, w, n, s}. If each letter in S is regarded as an integer lattice step, e = (1,0), w = (-1,0), n = (0,1), s = (0,-1), then each word is a path in the two-dimensional integer lattice starting from (0,0), never going below the x-axis and ending on the x-axis. Thus, this is a variant of Motzkin paths with two kinds of level steps. The algebraic definition is L = 1 + Le + Lw + LnLs - w where each word is regarded as a noncommutative monomial with variables in S. Replacing each letter in S by x and L by the g.f. A(x) leads to x + A(x) = (1 + x*A(x))^2. If we let y = x + x*x*A, then y^2 - y = x^3 - x which is an elliptic curve. - Michael Somos, Mar 28 2020

The Hankel number wall for the sequence L(0), L(1), ... has a zigzag diagonal sequence b(0) = 1, b(1) = 1. b(2) = e, b(3) = ew+ns, b(4) = na(ee-ew-ns), ... which is a generalized Somos-5 sequence with b(i)*b(i+5) = -n*s*b(i+1)*b(i+4) + e*n*s*b(i+2)*b(i+3). Define sequence c(0) = 0, c(1) = 1, c(i) = b(i-2) for i>1, and c(i) = -(-n*s)^(-i)*c(-i) if i<0. Then c(i)*c(i+5) = -n*s*c(i+1)*c(i+4) + e*n*s*c(i+2)*c(i+3) for all i in Z. If e=w=n=s=1, then c(i) = A006769(i) * (-1)^[mod(i,4)=3]. - Michael Somos, Oct 14 2024

			L(0) = 1, L(1) = e, L(2) = ee + ew + ns, L(3) = eee + ewe + nse + eew + eww + nsw + nes + ens.
G.f. = 1 + x + 3*x^2 + 8*x^3 + 23*x^4 + 68*x^5 + 207*x^6 + 644*x^7 + ...

Michael De Vlieger, Table of n, a(n) for n = 0..1000
Paul Barry, Integer sequences from elliptic curves, arXiv:2306.05025 [math.NT], 2023.
Hanna Mularczyk, Lattice Paths and Pattern-Avoiding Uniquely Sorted Permutations, arXiv:1908.04025 [math.CO], 2019.
Michael Somos, Number Walls in Combinatorics.

Cf. A001006, A006720, A006769, A025262.

CoefficientList[Series[(1 - 2 x - Sqrt[1 - 4 x + 4 x^3])/(2 x^2), {x, 0, 26}], x] (* Michael De Vlieger, Oct 30 2019 *)
a[ n_] := SeriesCoefficient[ (2 - 2*x)/(1 - 2*x + (1 - 4*x + 4*x^3)^(1/2)), {x, 0, n}]; (* Michael Somos, Oct 27 2024 *)
a[ n_] := If[ n<0, 0, SeriesCoefficient[Nest[(1 + x*#)^2 - x&, 1 + O[x], n], {x, 0, n}]]; (* Michael Somos, Oct 27 2024 *)

{a(n) = if( n<0, 0, polcoef( (1 - 2*x - sqrt( 1 - 4*x + 4*x^3 + x^3 * O(x^n)) ) / (2*x^2), n))};

{a(n) = if( n<0, 0, polcoef( (2 - 2*x)/(1 - 2*x + (1 - 4*x + 4*x^3 + x*O(x^n))^(1/2)), n))}; /* Michael Somos, Oct 27 2024 */

L = 1 + Le + Lw + LnLs - w.
a(n) = 2*a(n-1) + a(0)*a(n-2) + ... + a(n-2)*a(0) for n>1.
The Somos-4 sequence A006720(n+2) is the Hankel transform of a(n-1). See A001906 for definition of Hankel transform.
Let s(n)= A006769(n). Then 0 = f( -s(n-1) * s(n+1) / s(n)^2, -s(n) * s(n+2) / s(n+1)^2 ) where f(u, v) = u + v - (1 + u*v)^2.
G.f. A(x) satisfies 0 = f(x, A(x)) where f(u, v) = u + v - (1 + u*v)^2.
G.f.: (1 - 2*x - sqrt( 1 - 4*x + 4*x^3) ) / (2*x^2).
From Paul Barry, Mar 04 2010: (Start)
G.f.: ((1-x)/(1-2x))c(x^2(1-x)/(1-2x)^2), c(x) the g.f. of A000108;
a(n) = Sum_{k=0..floor(n/2)} (A000108(k) * Sum_{i=0..k+1} C(k+1,i)*C(n-i,n-2k-i)*(-1)^i*2^(n-2k-i)). (End)
a(n) = A025262(n+2) if n >= 0.
0 = a(n)*(+16*a(n+1) - 64*a(n+3) + 22*a(n+4)) + a(n+1)*(+32*a(n+2) - 14*a(n+3)) + a(n+2)*(+16*a(n+3) - 10*a(n+4)) + a(n+3)*(+2*a(n+3) + a(n+4)) if n>=0. - Michael Somos, Jan 18 2015

A050512 a(n) = (a(n-1)*a(n-3) - a(n-2)^2) / a(n-4), with a(0) = 0, a(1) = a(2) = a(3) = 1, a(4) = -1.

Original entry on oeis.org

0, 1, 1, 1, -1, -2, -3, -1, 7, 11, 20, -19, -87, -191, -197, 1018, 2681, 8191, -5841, -81289, -261080, -620551, 3033521, 14480129, 69664119, -2664458, -1612539083, -7758440129, -37029252553, 181003520899, 1721180313660, 12437589708389, 19206818781913

Offset: 0

Michael Somos, Dec 28 1999

From Paul Barry, May 31 2010: (Start)
a(n+1) is (-1)^binomial(n,2) times the Hankel transform of the sequence with g.f.
1/(1-x/(1+x^2/(1-x^2/(1-2x^2/(1+(3/4)x^2/(1+(2/9)x^2/(1+21)x^2/(1-... where
-1,1,2,-3/4,-2/9,21,... are the x-coordinates of the multiples of z=(0,0) on the elliptic curve E: y^2 - 2xy - y = x^3-x. (End)
This is a strong elliptic divisibility sequence t_n as given in [Kimberling, p. 16] where x = 1, y = 1, z = -1.
The elliptic curve y^2 + y = x^3 + x^2 (LMFDB label 43.a1) has infinite order point P = (0, 0). The x coordinate of n*P has denominator a(n)^2. - Michael Somos, Feb 14 2023

			G.f. = x + x^2 + x^3 - x^4 - 2*x^5 - 3*x^6 - x^7 + 7*x^8 + 11*x^9 + 20*x^10 + ...

Reinhard Zumkeller, Table of n, a(n) for n = 0..250
Paul Barry, Jacobsthal Decompositions of Pascal's Triangle, Ternary Trees, and Alternating Sign Matrices, Journal of Integer Sequences, 19, 2016, #16.3.5.
Paul Barry, Riordan Pseudo-Involutions, Continued Fractions and Somos 4 Sequences, arXiv:1807.05794 [math.CO], 2018.
Clark Kimberling, Strong divisibility sequences and some conjectures, Fib. Quart., 17 (1979), 13-17.
LMFDB, Elliptic Curve 43.a1 (Cremona label 43a1)

Cf. A006769.

a050512 n = a050512_list !! n
a050512_list = 0 : 1 : 1 : 1 : (-1) : zipWith div (zipWith (-) (zipWith (*)
   (drop 4 a050512_list) (drop 2 a050512_list))
     (map (^ 2) (drop 3 a050512_list))) (tail a050512_list)
-- Reinhard Zumkeller, Nov 02 2011

a[n_?OddQ] := a[n] = a[(n-1)/2]^3*a[(n+3)/2] - a[(n-3)/2]*a[(n+1)/2]^3; a[n_?EvenQ] := a[n] = (a[n/2-1]^2*a[n/2+2] - a[n/2-2]*a[n/2+1]^2)*a[n/2]; a[0] = 0; a[1] = a[2] = a[3] = 1; a[4] = -1; Table[a[n], {n, 0, 30}] (* Jean-François Alcover, Nov 29 2011 *)
Join[{0},RecurrenceTable[{a[1]==a[2]==a[3]==1,a[4]==-1,a[n]==(a[n-1] a[n-3]-a[n-2]^2)/a[n-4]},a,{n,30}]] (* Harvey P. Dale, Mar 23 2012 *)

an=vector(200); for(n=1,4,an[ n ]=[ 1,1,1,-1 ][ n ]); for(n=5, length(an),an[ n ]=(an[ n-1 ]*an[ n-3 ]-an[ n-2 ]^2)/an[ n-4 ]); a(n) =sign(n)*an[ abs(n)+(n==0) ]

{a(n) = my(an); if( n<0, -a(-n), if( n==0, 0, an = vector( max(4, n), i, 1); an[4] = -1; for( k=5, n, an[k] = (an[k-1] * an[k-3] - an[k-2]^2) / an[k-4]); an[n]))}; /* Michael Somos, Jul 07 2014 */

a(2*n + 1) = a(n + 2) * a(n)^3 - a(n - 1) * a(n + 1)^3 for all n in Z.
a(2*n) = a(n + 2) * a(n) * a(n - 1)^2 - a(n) * a(n - 2) * a(n + 1)^2 for all n in Z.
0 = a(n)*a(n+5) - a(n+1)*a(n+4) - a(n+2)*a(n+3) for all n in Z. - Michael Somos, Jul 07 2014
0 = a(n)*a(n+6) + a(n+1)*a(n+5) - 2*a(n+2)*a(n+4) for all n in Z. - Michael Somos, Jul 07 2014
a(n) = -a(-n) for all n in Z. - Michael Somos, Feb 14 2023

A247369 a(n) = (a(n-1) * a(n-3) + (-1)^n * a(n-2)^2) / a(n-4), with a(0) = 0, a(1) = -1, a(2) = a(3) = a(4) = 1, a(9) = 3.

Original entry on oeis.org

0, -1, 1, 1, 1, 0, 1, 1, 1, 3, 4, -5, 1, -7, 9, 8, 25, -23, 49, 87, 16, 295, 529, -903, 841, -1256, 3481, -1495, 16641, -44341, 98596, 217651, 4225, 1058961, 2337841, -5106896, 13608721, 5415345, 67387681, -173830481, 264517696, -2288275633, 6941055969

Offset: 0

Michael Somos, Sep 14 2014

sign

Reinhard Zumkeller, Table of n, a(n) for n = 0..400

Cf. A006769.

a247369 n = a247369_list !! n
a247369_list = [0, -1, 1, 1, 1, 0] ++ xs where
   xs = [1, 1, 1, 3] ++ zipWith (flip div) xs (zipWith (+)
              (zipWith (*) (tail xs) (drop 3 xs))
              (zipWith (*) (cycle [1, -1]) (map (^ 2) $ drop 2 xs)))
-- Reinhard Zumkeller, Sep 15 2014

I:=[3, 4, -5, 1]; [0, -1, 1, 1, 1, 0, 1, 1, 1] cat [n le 4 select I[n] else ( Self(n-1)*Self(n-3) + (-1)^n*Self(n-2)^2 )/Self(n-4): n in [1..30]]; // G. C. Greubel, Aug 05 2018

Join[{0, -1, 1, 1, 1, 0, 1, 1, 1}, RecurrenceTable[{a[9]==3, a[10]==4, a[11]==-5, a[12]==1, a[n]==(a[n-1]a[n-3] + (-1)^n a[n-2]^2)/a[n-4]}, a, {n, 9, 30}]] (* G. C. Greubel, Aug 05 2018 *)

{a(n) = my(A = [-1, 1, 1, 1]); n=abs(n); if( n==0, 0, if( n<5, A[n], A = concat(A, vector(n-4)); for(k=5, n, A[k] = if( k==9, 3, (A[k-1] * A[k-3] + (-1)^k * A[k-2]^2) / A[k-4])); A[n]))};

0 = a(n)*a(n+9) + a(n+1)*a(n+8) + a(n+3)*a(n+6) + a(n+4)*a(n+5) for all n in Z.

a(n) = a(-n), a(2*n) = A006769(n)^2 for all n in Z.

cp's OEIS Frontend

A278314 a(n) = -c(n-1) * c(n-2) * c(n+3) where c(n) = A006769(n).

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A178072 Sequence with Hankel transform equal to the Somos-4 sequence A006769(n+2).

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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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A025262 a(n) = a(1)*a(n-1) + a(2)*a(n-2) + ... + a(n-1)*a(1) for n >= 4.

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A028940 a(n) = numerator of the X-coordinate of n*P where P is the generator [0,0] for rational points on the curve y^2 + y = x^3 - x.

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

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A028941 Denominator of X-coordinate of n*P where P is the generator [0,0] for rational points on curve y^2+y = x^3-x.

A025262 a(n) = a(1)a(n-1) + a(2)a(n-2) + ... + a(n-1)*a(1) for n >= 4.