OFFSET
0,5
COMMENTS
This is a strong divisibility sequence; that is, if n divides m, then a(n) divides a(m) and moreover for all positive integer n,m a(gcd(n, m)) = gcd(a(n), a(m)).
This is a strong elliptic divisibility sequence t_n as given in [Kimberling, p. 16] where x = 1, y = -1, z = -5. - Michael Somos, Jul 07 2014
The elliptic curve y^2 + y = x^3 - x has LMFDB label 37.a1 (Cremona label 37a1). - Michael Somos, Feb 07 2024
LINKS
T. D. Noe, Table of n, a(n) for n = 0..100
Clark Kimberling, Strong divisibility sequences and some conjectures, Fib. Quart., 17 (1979), 13-17.
R. W. Gosper and Richard C. Schroeppel, Somos Sequence Near-Addition Formulas and Modular Theta Functions, arXiv:math/0703470 [math.NT], 2007.
LMFDB, Elliptic Curve 37.a1
Helmut Ruhland, Somos-4 and a quartic Surface in RP^3, arXiv:2312.02085 [math.AG], 2023.
FORMULA
a(n) = (a(n-1) * a(n-3) + a(n-2)^2) / a(n-4).
a(n) = (-a(n-1) * a(n-4) + 5 * a(n-2) * a(n-3)) / a(n-5).
a(2*n + 1) = a(n+2) * a(n)^3 - a(n-1) * a(n+1)^3.
a(2*n) = a(n+2) * a(n) * a(n-1)^2 - a(n) * a(n-2) * a(n+1)^2.
0 = a(n)*a(n+4) - a(n+1)*a(n+3) - a(n+2)*a(n+2) for all n in Z. - Michael Somos, Jul 07 2014
0 = a(n)*a(n+5) + a(n+1)*a(n+4) - 5*a(n+2)*a(n+3) for all n in Z. - Michael Somos, Jul 07 2014
EXAMPLE
G.f. = x + x^2 - x^3 - 5*x^4 - 4*x^5 + 29*x^6 + 129*x^7 - 65*x^8 + ...
MATHEMATICA
a[n_ /; n < 0] := -a[-n]; a[0] = 0; A006769[n_] := (ClearAll[an]; an[_] = 1; an[3] = -1; For[k = 5, k <= n, k++, an[k] = (an[k-1]*an[k-3] + an[k-2]^2)/an[k-4]]; an[n]); a[n_] := A006769[2n]; Table[a[n], {n, 0, 21}] (* Jean-François Alcover, Apr 11 2012, after 2nd formula *)
PROG
(PARI) an = vector(200); an = concat([ 1, 1, -1, -5 ], an); for( n=5, length(an), an[ n ]=(an[ n-1 ] * an[ n-3 ] + an[ n-2 ]^2) / an[ n-4 ]); a(n) = an[ n ]
(PARI) {a(n) = my(v = [1, 1, -1, -5]); if( n<0, -a(-n), if( n==0, 0, if( n<5, v[n], v = concat( v, vector(n - 4)); for( k=5, n, v[k] = (v[k-1] * v[k-3] + v[k-2]^2) / v[k-4]); v[n])))}; /* Michael Somos, Feb 12 2012 */
CROSSREFS
KEYWORD
sign,easy,nice
AUTHOR
Michael Somos, Oct 12 1999
STATUS
approved