|
| |
|
|
A006721
|
|
Somos-5 sequence: a(n) = (a(n-1) * a(n-4) + a(n-2) * a(n-3)) / a(n-5), a(0) = a(1) = a(2) = a(3) = 1.
(Formerly M0735)
|
|
12
| |
|
|
1, 1, 1, 1, 1, 2, 3, 5, 11, 37, 83, 274, 1217, 6161, 22833, 165713, 1249441, 9434290, 68570323, 1013908933, 11548470571, 142844426789, 2279343327171, 57760865728994, 979023970244321, 23510036246274433, 771025645214210753
(list; graph; refs; listen; history; internal format)
|
|
|
|
OFFSET
| 0,6
|
|
|
COMMENTS
| Using the addition formula for the Weierstrass sigma function it is simple to prove that the subsequence of even terms of a Somos-5 type sequence satisfy a 4th order recurrence of Somos-4 type and similarly the odd subsequence satsifies the same 4th order recurrence. - Andrew Hone (anwh(AT)kent.ac.uk), Aug 24 2004
log(a(n)) ~ 0.071626946 * n^2. (Hone)
The Brown link article gives interesting information about related sequences including recurrences and numerical approximations.
|
|
|
REFERENCES
| R. H. Buchholz and R. L. Rathbun, "An infinite set of Heron triangles with two rational medians", Amer. Math. Monthly, 104 (1997), 107-115.
David Gale, "The strange and surprising saga of the Somos sequences", Math. Intelligencer 13(1) (1991), pp. 40-42.
A. N. W. Hone, Elliptic curves and quadratic recurrence sequences, Bull. Lond. Math. Soc. 37 (2005) 161-171.
J. L. Malouf, "An integer sequence from a rational recursion", Discr. Math. 110 (1992), 257-261.
R. M. Robinson, "Periodicity of Somos sequences", Proc. Amer. Math. Soc., 116 (1992), 613-619.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
Alfred J. van der Poorten, Elliptic Curves and Continued Fractions, Journal of Integer Sequences, Vol. 8 (2005), Article 05.2.5.
|
|
|
LINKS
| T. D. Noe, Table of n, a(n) for n=0..100
K. S. Brown, A Quasi-Periodic Sequence
S. Fomin and A. Zelevinsky, The Laurent phenomemon
A. N. W. Hone, Sigma function solution of the initial value problem for Somos 5 sequences
J. Propp, The Somos Sequence Site
J. Propp, The 2002 REACH tee-shirt
M. Somos, Somos 6 Sequence
M. Somos, Brief history of the Somos sequence problem
D. E. Speyer, Perfect matchings and the octahedral recurrence
A. J. van der Poorten, Elliptic curves and continued fractions
A. J. van der Poorten, Recurrence relations for elliptic sequences...
A. J. van der Poorten, Hyperelliptic curves, continued fractions and Somos sequences
Eric Weisstein's World of Mathematics, Link to a section of The World of Mathematics.
D. Zagier, Problems posed at the St Andrews Colloquium, 1996
Index entries for two-way infinite sequences
|
|
|
FORMULA
| Comments from Andrew Hone (anwh(AT)kent.ac.uk), Aug 24 2004: "Both the even terms b(n)=a(2n) and odd terms b(n)=a(2n+1) satisfy the fourth order recurrence b(n)=(b(n-1)*b(n-3)+8*b(n-2)^2)/b(n-4).
"Hence the general formula is a(2n)=A*B^n*sigma(c+n*k)/sigma(k)^(n^2), a(2n+1)=D*E^n*sigma(f+n*k)/sigma(k)^(n^2) where sigma is the Weierstrass sigma function associated to the elliptic curve y^2=4*x^3-(121/12)*x+845/216 (this is birationally equivalent to the minimal model V^2+U*V+6*V=U^3+7*U^2+12*U given by van der Poorten).
"The real/imaginary half-periods of the curve are w1=1.181965956, w3=0.973928783*I and the constants are A=0.142427718-1.037985022*I, B=0.341936209+0.389300717*I, c=0.163392411+w3, k=1.018573545, D=-0.363554228-0.803200610*I, E=0.644801269+0.734118205*I, f=c+k/2-w1 all to 9 decimal places."
a(4 - n) = a(n). a(n+2) * a(n-2) = 2 * a(n+1) * a(n-1) - a(n)^2 if n is even. a(n+2) * a(n-2) = 3 * a(n+1) * a(n-1) - a(n)^2 if n is odd.
|
|
|
MATHEMATICA
| a[0] = a[1] = a[2] = a[3] = a[4] = 1; a[n_] := a[n] = (a[n - 1] a[n - 4] + a[n - 2] a[n - 3])/a[n - 5]; Array[a, 27, 0] [From Robert G. Wilson v (rgwv(AT)rgwv.com), Aug 15 2010]
a[ n_] := If[ Abs [n - 2] < 3, 1, If[ n < 0, a[4 - n], a[n] = (a[n - 1] a[n - 4] + a[n - 2] a[n - 3]) / a[n - 5]]] (* Michael Somos, Jul 15 2011 *)
RecurrenceTable[{a[0]==a[1]==a[2]==a[3]==a[4]==1, a[n]==(a[n-1]a[n-4]+ a[n-2]a[n-3])/a[n-5]}, a, {n, 30}] (* From Harvey P. Dale, Dec 25 2011 *)
|
|
|
PROG
| (PARI) {a(n) = if( abs(n-2) < 3, 1, if( n<0, a(4-n), (a(n-1) * a(n-4) + a(n-2) * a(n-3)) / a(n-5)))} /* Michael Somos, Jul 15 2011 */
(Haskell)
a006721 n = a006721_list !! n
a006721_list = [1, 1, 1, 1, 1] ++
zipWith div (foldr1 (zipWith (+)) (map b [1..2])) a006721_list
where b i = zipWith (*) (drop i a006721_list) (drop (5-i) a006721_list)
-- Reinhard Zumkeller, Jan 22 2012
|
|
|
CROSSREFS
| Cf. A006720, A006722, A006723, A048736.
Sequence in context: A124561 A167604 A065510 * A111289 A127181 A113734
Adjacent sequences: A006718 A006719 A006720 * A006722 A006723 A006724
|
|
|
KEYWORD
| easy,nonn,nice
|
|
|
AUTHOR
| N. J. A. Sloane (njas(AT)research.att.com).
|
|
|
EXTENSIONS
| a(26) & a(27) from Robert G. Wilson v (rgwv(AT)rgwv.com), Aug 15 2010
|
| |
|
|
