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%I M2457 #98 Feb 02 2024 07:46:47
%S 1,1,1,1,1,1,3,5,9,23,75,421,1103,5047,41783,281527,2534423,14161887,
%T 232663909,3988834875,45788778247,805144998681,14980361322965,
%U 620933643034787,16379818848380849,369622905371172929,20278641689337631649,995586066665500470689
%N 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.
%D C. Pickover, Mazes for the Mind, St. Martin's Press, NY, 1992, p. 350.
%D N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%H T. D. Noe, <a href="/A006722/b006722.txt">Table of n, a(n) for n=0..100</a>
%H R. H. Buchholz and R. L. Rathbun, <a href="http://www.jstor.org/stable/2974977">An infinite set of Heron triangles with two rational medians</a>, Amer. Math. Monthly, 104 (1997), 107-115.
%H Chang, Xiangke; Hu, Xingbiao, <a href="https://doi.org/10.1016/j.laa.2012.01.016">A conjecture based on Somos-4 sequence and its extension</a>, Linear Algebra Appl. 436, No. 11, 4285-4295 (2012).
%H Yuri N. Fedorov and Andrew N. W. Hone, <a href="http://arxiv.org/abs/1512.00056">Sigma-function solution to the general Somos-6 recurrence via hyperelliptic Prym varieties</a>, arXiv:1512.00056 [nlin.SI], 2015.
%H S. Fomin and A. Zelevinsky, <a href="http://arXiv.org/abs/math.CO/0104241">The Laurent phenomenon</a>, arXiv:math/0104241 [math.CO], 2001.
%H David Gale, <a href="http://dx.doi.org/10.1007/BF03024070">The strange and surprising saga of the Somos sequences</a>, Math. Intelligencer 13(1) (1991), pp. 40-42.
%H R. W. Gosper and Richard C. Schroeppel, <a href="http://arxiv.org/abs/math/0703470">Somos Sequence Near-Addition Formulas and Modular Theta Functions</a>, arXiv:math/0703470 [math.NT], 2007.
%H A. N. W. Hone, <a href="http://dx.doi.org/10.1080/00036810903329977">Analytic solutions and integrability for bilinear recurrences of order six</a>, Appl. Anal. 89, no.4 (2010), 473-492.
%H J. L. Malouf, <a href="http://dx.doi.org/10.1016/0012-365X(92)90714-Q">An integer sequence from a rational recursion</a>, Discr. Math. 110 (1992), 257-261.
%H R. M. Robinson, <a href="http://dx.doi.org/10.1090/S0002-9939-1992-1140672-5">Periodicity of Somos sequences</a>, Proc. Amer. Math. Soc., 116 (1992), 613-619.
%H Vladimir Shevelev and Peter J. C. Moses, <a href="http://arxiv.org/abs/1112.5715">On a sequence of polynomials with hypothetically integer coefficients</a>, arXiv preprint arXiv:1112.5715 [math.NT], 2011.
%H Michael Somos, <a href="http://grail.eecs.csuohio.edu/~somos/somos6.html">Somos 6 Sequence</a>
%H Michael Somos, <a href="http://faculty.uml.edu/jpropp/somos/history.txt">Brief history of the Somos sequence problem</a>
%H Alex Stone, <a href="https://www.quantamagazine.org/the-astonishing-behavior-of-recursive-sequences-20231116/">The Astonishing Behavior of Recursive Sequences</a>, Quanta Magazine, Nov 16 2023, 13 pages.
%H A. van der Poorten, <a href="http://arXiv.org/abs/math.NT/0608247">Hyperelliptic curves, continued fractions and Somos sequences</a>, arXiv:math/0608247 [math.NT], 2006.
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/SomosSequence.html">Somos Sequence.</a>
%H <a href="/index/Tu#2wis">Index entries for two-way infinite sequences</a>
%F a(n) = a(5-n).
%F 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)
%F 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:
%F Fix the following seven constants:
%F c1 = 0.875782749065950194217251...,
%F c2 = 1.084125925473763343779968...,
%F c3 = 0.114986002186402203509006...,
%F c4 = 0.077115634258697284328024...,
%F c5 = 1.180397390176742642553759...,
%F c6 = 1.508030831265086447098989..., and
%F c7 = 2.551548771413081602906643... .
%F 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.
%F 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
%F From _Andrew Hone_ and Yuri Fedorov, Nov 27 2015: (Start)
%F The following is an exact formula for a(n):
%F a(n+3) = A*B^n*C^(n^2 -1)*sigma(v_0 + n*v) / sigma(v)^(n^2),
%F where
%F A = C / sigma(v_0),
%F B = A^(-1)*sigma(v) / sigma(v_0+v),
%F C = i/sqrt(20) (with i the imaginary unit),
%F 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
%F 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
%F P_1 = ( -8 + sqrt(65), 20*i*(129 -16*sqrt(65)) ),
%F P_2 = ( -8 - sqrt(65), 20*i*(129 +16*sqrt(65)) ),
%F Q_1 = ( 5 + 2*sqrt(6), 4*i*(71 +sqrt(6)) ),
%F Q_2 = ( 5 - 2*sqrt{6}, 4*i*(71 -sqrt(6)) ).
%F The Abel map is based at infinity and calculated with respect to the basis of holomorphic differentials dx/y, x dx/y.
%F 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).
%F (End)
%t 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 *)
%t 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 *)
%o (PARI) {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 */
%o (Haskell)
%o a006722 n = a006722_list !! n
%o a006722_list = [1,1,1,1,1,1] ++
%o zipWith div (foldr1 (zipWith (+)) (map b [1..3])) a006722_list
%o where b i = zipWith (*) (drop i a006722_list) (drop (6-i) a006722_list)
%o -- _Reinhard Zumkeller_, Jan 22 2012
%o (Python)
%o from gmpy2 import divexact
%o A006722 = [1,1,1,1,1,1]
%o for n in range(6,101):
%o ....A006722.append(divexact(A006722[n-1]*A006722[n-5]+A006722[n-2]*A006722[n-4]+A006722[n-3]**2,A006722[n-6]))
%o # _Chai Wah Wu_, Sep 01 2014
%o (Magma) [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
%Y Cf. A006720, A006721, A006723, A048736.
%K nonn,easy,nice
%O 0,7
%A _N. J. A. Sloane_
%E More terms from _James A. Sellers_, Aug 22 2000
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