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A006721
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Somos-5 sequence: a(n) = (a(n-1) * a(n-4) + a(n-2) * a(n-3)) / a(n-5), with a(0) = a(1) = a(2) = a(3) = a(4) = 1.
(Formerly M0735)
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33
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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
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history;
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OFFSET
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0,6
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COMMENTS
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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 satisfies the same 4th-order recurrence. - Andrew Hone, 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.
The n-th term is a divisor of the (n+k*(2*n-4))-th term for all integers n and k. - Peter H van der Kamp, May 18 2015
The elliptic curve y^2 + xy = x^3 + x^2 - 2x (LMFDB label 102.a1) has infinite order point P = (2, 2) and 2-torsion point T = (0, 0). Define d(n) = a(n+2). The x and y coordinates of nP + T have denominators d(n)^2 and d(n)^3. - Michael Somos, Oct 29 2022
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REFERENCES
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Paul C. Kainen, Fibonacci in Somos-5 ..., Fib. Q., 60:4 (2022), 362-364.
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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Comments from Andrew Hone, 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.
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MAPLE
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for n from 0 to 4 do a[n]:= 1 od:
for n from 5 to 50 do a[n]:=(a[n-1] * a[n-4] + a[n-2] * a[n-3]) / a[n-5] od:
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MATHEMATICA
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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] (* Robert G. Wilson v, 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}] (* Harvey P. Dale, Dec 25 2011 *)
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PROG
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(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 */
(PARI) {a(n) = my(E = ellinit([1, 1, 0, -2, 0]), P = [2, 2], T = [0, 0]); if(n == 2, 1, n = abs(n-2); sqrtint(denominator(elladd(E, T, ellmul(E, P, n))[1])))}; /* Michael Somos, Oct 29 2022 */
(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)
(Python)
from gmpy2 import divexact
for n in range(5, 1001):
(Magma) I:=[1, 1, 1, 1, 1]; [n le 5 select I[n] else (Self(n-1) * Self(n-4) + Self(n-2) * Self(n-3)) div Self(n-5): n in [1..30]]; // Vincenzo Librandi, May 18 2015
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CROSSREFS
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KEYWORD
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easy,nonn,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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