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A006722
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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.
(Formerly M2457)
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16
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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
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OFFSET
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0,7
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REFERENCES
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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).
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LINKS
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FORMULA
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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)
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MATHEMATICA
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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 *)
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PROG
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(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 */
(Haskell)
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)
(Python)
from gmpy2 import divexact
for n in range(6, 101):
(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
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CROSSREFS
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KEYWORD
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nonn,easy,nice
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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