A006723 - OEIS

%I M2456 #84 Feb 02 2024 07:48:00

%S 1,1,1,1,1,1,1,3,5,9,17,41,137,769,1925,7203,34081,227321,1737001,

%T 14736001,63232441,702617001,8873580481,122337693603,1705473647525,

%U 22511386506929,251582370867257,9254211194697641,215321535159114017

%N 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.

%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="/A006723/b006723.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 S. Fomin and A. Zelevinsky, <a href="https://arxiv.org/abs/math/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="https://arxiv.org/abs/math/0703470">Somos Sequence Near-Addition Formulas and Modular Theta Functions</a>, arXiv:math/0703470 [math.NT], 2007.

%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 J. Propp, <a href="http://faculty.uml.edu/jpropp/somos.html">The Somos Sequence Site</a>

%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="https://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/somos7.html">Somos 7 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="https://arxiv.org/abs/math/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(6 - n) = a(n) for all n in Z.

%F a(n) = ((8-2*(-1)^n)*a(n-5)*a(n-3)-a(n-4)^2)/a(n-8). - _Bruno Langlois_, Aug 09 2016

%t 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 *)

%o (PARI)  {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])};

%o (Haskell)

%o a006723 n = a006723_list !! n

%o a006723_list = [1,1,1,1,1,1,1] ++

%o   zipWith div (foldr1 (zipWith (+)) (map b [1..3])) a006723_list

%o   where b i = zipWith (*) (drop i a006723_list) (drop (7-i) a006723_list)

%o -- _Reinhard Zumkeller_, Jan 22 2012

%o (Python)

%o from gmpy2 import divexact

%o A006723 = [1,1,1,1,1,1,1]

%o for n in range(7,101):

%o ....A006723.append(divexact(A006723[n-1]*A006723[n-6]+A006723[n-2]*A006723[n-5]+A006723[n-3]*A006723[n-4],A006723[n-7]))

%o # _Chai Wah Wu_, Sep 01 2014

%o (Magma) 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

%Y Cf. A006720, A006721, A006722, A048736.

%Y Cf. A078495.

%K nonn,easy,nice

%O 0,8

%A _N. J. A. Sloane_

%E More terms from _James A. Sellers_, Aug 22 2000