A051138 -id:A051138 - cp's OEIS frontend

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

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

Offset: 0

N. J. A. Sloane

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

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

Robert G. Wilson v, Table of a(n) for n = 0..100.
Paul Barry, Riordan-Bernstein Polynomials, Hankel Transforms and Somos Sequences, Journal of Integer Sequences, Vol. 15 2012, #12.8.2. - From _N. J. A. Sloane_, Dec 29 2012
Paul Barry, On the Hurwitz Transform of Sequences, Journal of Integer Sequences, Vol. 15 (2012), #12.8.7.
Paul Barry, Riordan arrays, the A-matrix, and Somos 4 sequences, arXiv:1912.01126 [math.CO], 2019.
Paul Barry, Integer sequences from elliptic curves, arXiv:2306.05025 [math.NT], 2023.
H. W. Braden, V. Z. Enolskii and A. N. W. Hone, Bilinear recurrences and addition formulas for hyperelliptic sigma functions, arXiv:math/0501162 [math.NT], 2005.
R. H. Buchholz and R. L. Rathbun, An infinite set of Heron triangles with two rational medians, Amer. Math. Monthly, 104 (1997), 107-115.
Xiangke Chang and Xingbiao Hu, A conjecture based on Somos-4 sequence and its extension, Linear Algebra Appl. 436, No. 11, 4285-4295 (2012).
Harini Desiraju and Brady Haran, The Troublemaker Number, Numberphile video (2022).
S. B. Ekhad and D. Zeilberger, How To Generate As Many Somos-Like Miracles as You Wish, arXiv preprint arXiv:1303.5306[math.CO], 2013.
Graham Everest, Gerard Mclaren and Tom Ward, Primitive divisors of elliptic divisibility sequences, arXiv:math/0409540 [math.NT], 2004-2006.
G. Everest, S. Stevens, D. Tamsett and T. Ward, Primitive divisors of quadratic polynomial sequences, arXiv:math/0412079v1 [math.NT], 2004.
G. Everest et al., Primes generated by recurrence sequences, arXiv:math/0412079 [math.NT], 2006.
G. Everest et al., Primes generated by recurrence sequences, Amer. Math. Monthly, 114 (No. 5, 2007), 417-431.
S. Fomin and A. Zelevinsky, The Laurent phenomenon, arXiv:math/0104241 [math.CO], 2001.
Allan Fordy and Andrew Hone, Discrete integrable systems and Poisson algebras from cluster maps, arXiv preprint arXiv:1207.6072 [nlin.SI], 2012.
A. P. Fordy, Periodic Cluster Mutations and Related Integrable Maps, arXiv preprint arXiv:1403.8061 [math-ph], 2014.
A. P. Fordy, Mutation-periodic quivers, integrable maps and associated Poisson algebras, Phil Trans. R. Soc. Lond. Ser A (Math. Phys. Eng. Sci.) 369 (1939) (2011) 1264-1279.
David Gale, The strange and surprising saga of the Somos sequences, in Mathematical Entertainments, Math. Intelligencer 13(1) (1991), pp. 40-42.
R. W. Gosper and Richard C. Schroeppel, Somos Sequence Near-Addition Formulas and Modular Theta Functions, arXiv:math/0703470 [math.NT], 2007.
A. N. W. Hone, Sigma function solution of the initial value problem for Somos 5 sequences, arXiv:math/0501554 [math.NT], 2005-2006.
A. N. W. Hone, Algebraic curves, integer sequences and a discrete Painlevé transcendent, Proceedings of SIDE 6, Helsinki, Finland, 2004; arXiv:0807.2538 [nlin.SI], 2008.
A. N. W. Hone, Elliptic curves and quadratic recurrence sequences, Bull. Lond. Math. Soc. 37 (2005) 161-171.
Andrew N. W. Hone, Growth of Mahler measure and algebraic entropy of dynamics with the Laurent property, arXiv:2109.08217 [math.NT], 2021.
A. N. W. Hone and R. Inoue, Discrete Painlevé equations from Y-systems, arXiv preprint arXiv:1405.5379 [math-ph], 2014
R. Jones and J. Rouse, Galois Theory of Iterated Endomorphisms, arXiv:0706.2384 [math.NT], 2007-2009; Proceedings of the London Mathematical Society, 100, no. 3 (2010), 763-794.
Xinrong Ma, Magic determinants of Somos sequences and theta functions, Discrete Mathematics 310.1 (2010): 1-5.
J. L. Malouf, An integer sequence from a rational recursion, Discr. Math. 110 (1992), 257-261.
Valentin Ovsienko and Serge Tabachnikov, Dual numbers, weighted quivers, and extended Somos and Gale-Robinson sequences, arXiv:1705.01623 [math.CO], 2017. See p. 3.
Kevin I. Piterman and Leandro Vendramin, Computer algebra with GAP, 2023. See p. 39.
J. Propp, The Somos Sequence Site
J. Propp, The 2002 REACH tee-shirt
R. M. Robinson, Periodicity of Somos sequences, Proc. Amer. Math. Soc., 116 (1992), 613-619.
Helmut Ruhland, Somos-4 and a quartic Surface in RP^3, arXiv:2312.02085 [math.AG], 2023.
Matthew Christopher Russell, Using experimental mathematics to conjecture and prove theorems in the theory of partitions and commutative and non-commutative recurrences, PhD Dissertation, Mathematics Department, Rutgers University, May 2016.
Vladimir Shevelev and Peter J. C. Moses, On a sequence of polynomials with hypothetically integer coefficients, arXiv preprint arXiv:1112.5715 [math.NT], 2011.
Michael Somos, Somos 6 Sequence
Michael Somos, Brief history of the Somos sequence problem
Michael Somos, Four polynomial sequences w,x,y,z are discrete versions of the four Jacobi theta functions or the four Weierstrass sigma functions, 2016.
D. E. Speyer, Perfect matchings and the octahedral recurrence, arXiv:math/0402452 [math.CO], 2004.
Alex Stone, The Astonishing Behavior of Recursive Sequences, Quanta Magazine, Nov 16 2023, 13 pages.
Andrei K. Svinin, Somos-4 equation and related equations, arXiv:2307.05866 [math.CA], 2023.
P. H. van der Kamp, Somos-4 and Somos-5 are arithmetic divisibility sequences, arXiv:1505.00194 [math.NT], 2015.
A. J. van der Poorten, Recurrence relations for elliptic sequences: every Somos 4 is a Somos k, arXiv:math/0412293 [math.NT], 2004.
A. J. van der Poorten, Hyperelliptic curves, continued fractions and Somos sequences, arXiv:math/0608247 [math.NT], 2006.
A. J. van der Poorten, Elliptic curves and continued fractions, J. Int. Sequences, Volume 8, no. 2 (2005), article 05.2.5.
Leandro Vendramin, Mini-couse on GAP - Exercises, Universidad de Buenos Aires (Argentina, 2020).
Eric Weisstein's World of Mathematics, Somos Sequence
Index entries for two-way infinite sequences

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

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

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

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

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

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

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

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

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

A028937 Denominator of x-coordinate of (2n)*P where P = (0,0) is the generator for rational points on the curve y^2 + y = x^3 - x.

Original entry on oeis.org

1, 1, 1, 25, 16, 841, 16641, 4225, 13608721, 264517696, 12925188721, 5677664356225, 49020596163841, 158432514799144041, 62586636021357187216, 1870098771536627436025, 41998153797159031581158401, 15402543997324146892198790401

Offset: 1

N. J. A. Sloane

			a(4) = 25 where 8P = (21/25, -69/125).

Seiichi Manyama, Table of n, a(n) for n = 1..106
LMFDB, Elliptic Curve 37.a1 (Cremona label 37a1)
B. Mazur, Arithmetic on curves, Bull. Amer. Math. Soc. 14 (1986), 207-259; see p. 225.

Cf. A006720, A051138, A028936 (numerator), A028938, A028939, A028941.

a(n)=denominator(ellmul(E,[0,0],2*n)[1]) \\ Charles R Greathouse IV, Mar 23 2022

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) = A028941(2n). - Seiichi Manyama, Nov 19 2016
a(n) = a(-n) = b(n)*b(n+3) - b(n+1)*b(n+2) for all n in Z where b(n) = A006720(n). - Michael Somos, Mar 23 2022

A157101 A Somos-4 variant.

Original entry on oeis.org

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

Offset: 0

Paul Barry, Feb 22 2009

Hankel transform of A157100.

G. C. Greubel, Table of n, a(n) for n = 0..145
Paul Barry, Generalized Catalan recurrences, Riordan arrays, elliptic curves, and orthogonal polynomials, arXiv:1910.00875 [math.CO], 2019.

Cf. A051138.

Cf. A157005, A162546, A162547.

a:=[1,-1,-5,-4];; for n in [5..20] do a[n]:=(a[n-1]*a[n-3] + a[n-2]^2)/a[n-4]; od; a; # G. C. Greubel, Feb 23 2019

I:=[1,-1,-5,-4]; [n le 4 select I[n] else (Self(n-1)*Self(n-3) + Self(n-2)^2)/Self(n-4): n in [1..20]]; // G. C. Greubel, Feb 23 2019

RecurrenceTable[{a[n]==(a[n-1]*a[n-3]+a[n-2]^2)/a[n-4], a[0]==1, a[1]==-1, a[2]==-5, a[3]==-4}, a, {n,20}] (* G. C. Greubel, Feb 23 2019 *)

m=20; v=concat([1,-1,-5,-4], vector(m-4)); for(n=5, m, v[n] = (v[n-1]*v[n-3] +v[n-2]^2)/v[n-4]); v \\ G. C. Greubel, Feb 23 2019

def a(n):
    if (n==0): return 1
    elif (n==1): return -1
    elif (n==2): return -5
    elif (n==3): return -4
    else: return (a(n-1)*a(n-3) + a(n-2)^2)/a(n-4)
[a(n) for n in (0..20)] # G. C. Greubel, Feb 23 2019

a(n) = (a(n-1)*a(n-3) + a(n-2)^2)/a(n-4), with a(0)=1, a(1)=-1, a(2)=-5, a(3)=-4.

a(n) = A051138(n+1) for all n in Z. - Michael Somos, Jul 17 2016

A157002 Transform of Catalan numbers whose Hankel transform gives the Somos-4 sequence.

Original entry on oeis.org

1, 0, 1, 2, 6, 17, 51, 156, 488, 1552, 5006, 16337, 53849, 179015, 599535, 2020924, 6851150, 23344138, 79902364, 274606264, 947240592, 3278404274, 11381240074, 39621423949, 138288477617, 483805404673, 1696318159457, 5959737806635

Offset: 0

Paul Barry, Feb 20 2009

Image of the Catalan numbers A000108 by the Riordan array (1-x,x(1-x^2)). Hankel transform is A006720(n+1).
The sequence a(n)+a(n+1) begins 1,1,3,8,23,68,... which is A056010. The sequence a(n)+a(n-1) begins 1,1,1,3,8,23,68,... which is A025262. This is obtained by applying (1-x^2,x(1-x^2)) to the Catalan numbers.
Hankel transform of a(n+1) is -A051138(n). - Michael Somos, Feb 10 2015

			G.f. = 1 + x^2 + 2*x^3 + 6*x^4 + 17*x^5 + 51*x^6 + 156*x^7 + 488*x^8 + ...

G. C. Greubel, Table of n, a(n) for n = 0..1000
Gouce Xin, Proof of the Somos-4 Hankel determinants conjecture, Advances in Applied Mathematics, Volume 42, Issue 2, February 2009, Pages 152-156.

Cf. A000108, A006720, A025262, A051138, A056010.

m:=30; R:=PowerSeriesRing(Rationals(), m); Coefficients(R!( (1 -Sqrt(1-4*x*(1-x^2)))/(2*x*(1+x)) )); // G. C. Greubel, Feb 26 2019

CoefficientList[Series[(1-Sqrt[1-4x(1-x^2)])/(2x(1+x)), {x,0,30}], x] (* G. C. Greubel, Feb 26 2019 *)

{a(n) = if( n<0, -(-1)^n / 2 * (n<-1), polcoeff( (1 - sqrt(1 - 4*x * (1 - x^2) + x^2 * O(x^n))) / (2 * x * (1 + x)), n))}; /* Michael Somos, Feb 10 2015 */

((1-sqrt(1-4*x*(1-x^2)))/(2*x*(1+x))).series(x, 30).coefficients(x, sparse=False) # G. C. Greubel, Feb 26 2019

G.f.: (1 - sqrt(1-4*x*(1-x^2)))/(2*x*(1+x)).
a(n) = Sum_{k=0..n} (-1)^floor((n-k+1)/2)*C(k,floor((n-k)/2))*A000108(k).
Conjecture: (n+1)*a(n) +3*(-n+1)*a(n-1) +2*(-2*n+1)*a(n-2) +2*(2*n-7)*a(n-3) +2*(2*n-7)*a(n-4)=0. - R. J. Mathar, Nov 19 2014
0 = a(n)*(+16*a(n+1) + 16*a(n+2) - 64*a(n+3) - 42*a(n+4) + 22*a(n+5)) + a(n+1)*(+16*a(n+1) + 48*a(n+2) - 46*a(n+3) - 56*a(n+4) + 22*a(n+5)) + a(n+2)*(+32*a(n+2) + 34*a(n+3) - 8*a(n+4) - 10*a(n+5)) + a(n+3)*(+18*a(n+3) + 11*a(n+4) - 9*a(n+5)) + a(n+4)*(+3*a(n+4) + a(n+5)) for all n in Z. - Michael Somos, Feb 10 2015

A171416 A sequence with Somos-4 Hankel transform.

Original entry on oeis.org

1, 1, 2, 3, 7, 13, 31, 65, 156, 351, 849, 1993, 4866, 11733, 28921, 70997, 176560, 438979, 1100302, 2761797, 6969909, 17625015, 44742636, 113822415, 290416803, 742486655, 1902767481, 4885201701, 12567065582, 32382099109, 83580301371

Offset: 0

Paul Barry, Dec 08 2009

Hankel transform is the Somos-4 sequence A006720(n+2).
The generating function A(x) satisfies A(x) = 1 + x + x^2*A(x) + (x*A(x))^2.
BINOMIAL transform is A087626. HANKEL transform with a(0) omitted is A051138(n+2). - Michael Somos, Jan 11 2013

			G.f. = 1 + x + 2*x^2 + 3*x^3 + 7*x^4 + 13*x^5 + 31*x^6 + 65*x^7 + 156*x^8 + ...

G. C. Greubel, Table of n, a(n) for n = 0..1000
Paul Barry, Riordan arrays, the A-matrix, and Somos 4 sequences, arXiv:1912.01126 [math.CO], 2019.

Cf. A006720, A051138, A087626, A377264.

R:=PowerSeriesRing(Rationals(), 30); Coefficients(R!((1-x^2-Sqrt(1-6*x^2-4*x^3+x^4))/(2*x^2))); // G. C. Greubel, Sep 22 2018

m:=30; S:=series((1 -x^2 -sqrt(1-6*x^2-4*x^3+x^4))/(2*x^2), x, m+1): seq(coeff(S, x, j), j=0..m); # G. C. Greubel, Feb 18 2020

CoefficientList[Series[(1-x^2 -Sqrt[1 -6x^2 -4x^3 +x^4])/(2x^2), {x, 0, 30}], x] (* Or *)
a[n_]:= a[n]= a[n-2] + Sum[a[k-1]a[n-k-1], {k, n-1}]; a[0]=a[1]=1; Array[a, 31, 0] (* Robert G. Wilson v, Mar 28 2011 *)

{a(n) = if( n<0, 0, polcoeff( 2*(1 + x) / (1 - x^2 + sqrt(1 - 6*x^2 - 4*x^3 + x^4 + x*O(x^n))), n))}; /* Michael Somos, Jan 11 2013 */

def A171416_list(prec):
    P. = PowerSeriesRing(ZZ, prec)
    return P( (1 -x^2 -sqrt(1-6*x^2-4*x^3+x^4))/(2*x^2) ).list()
A171416_list(30) # G. C. Greubel, Feb 18 2020

G.f.: (1 - x^2 - sqrt(1-6*x^2-4*x^3+x^4))/(2*x^2).
G.f.: (1/(1-x))*c(x^2/((1-x)*(1-x^2))) where c(x) is the g.f. of A000108.
G.f.: 1/(1-x-x^2/(1-x^2-x^2/(1-x-x^2/(1-x^2-x^2/(1-x-x^2/(1-x^2-x^2/(1-...))))))) (continued fraction).
a(n) = a(n-2) + Sum_{k=1..n-1} a(k-1)*a(n-k-1) with a(0)=a(1)=1.
Conjecture: (n+2)*a(n) +(n+1)*a(n-1) +6*(1-n)*a(n-2) +2*(11-5*n)*a(n-3) +(10-3*n)*a(n-4) +(n-5)*a(n-5)=0. - R. J. Mathar, Jul 24 2012
G.f.: 2*(1 + x) / (1 - x^2 + sqrt(1 - 6*x^2 - 4*x^3 + x^4)).
(n+2) * a(n) - (6*n-6) * a(n-2)  - (4*n-10) * a(n-3) + (n-4) * a(n-4) = 0 if n>3. - Michael Somos, Jan 11 2013
If we write the generating function as 1/(1-b_{0}*x/(1-c_{0}x/(1-b_{1}*x/(1-c_{1}*x/(1-...))))), then  b_{n}*c_{n} = A006720(n+1)*A006720(n+3)/A006720(n+2)^2 = A377264(n)/A006720(n+2)^2. - Thomas Scheuerle, Oct 22 2024

cp's OEIS Frontend

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

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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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A028937 Denominator of x-coordinate of (2n)*P where P = (0,0) is the generator for rational points on the curve y^2 + y = x^3 - x.

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A157101 A Somos-4 variant.

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A157002 Transform of Catalan numbers whose Hankel transform gives the Somos-4 sequence.

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A171416 A sequence with Somos-4 Hankel transform.

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