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A051138
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Divisibility sequence associated with elliptic curve y^2 + y = x^3 - x and point (1, 0).
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1
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0, 1, 1, -1, -5, -4, 29, 129, -65, -3689, -16264, 113689, 2382785, 7001471, -398035821, -7911171596, 43244638645, 6480598259201, 124106986093951, -5987117709349201, -541051130050800400, -4830209396684261199
(list; graph; refs; listen; history; internal format)
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OFFSET
| 0,5
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COMMENTS
| This is a divisibility sequence; that is, if n divides m, then a(n) divides a(m).
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REFERENCES
| C. Kimberling, Strong divisibility sequences and some conjectures, Fib. Quart., 17 (1979), 13-17.
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LINKS
| T. D. Noe, Table of n, a(n) for n=0..100
Index to divisibility sequences
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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.
a(-n) = -a(n). a(n) = A006769(2*n). a(n)^2 = A028937(n). |a(n)|^3 = A028939(n).
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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) = local(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 */
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CROSSREFS
| Cf. A006769, A006720, A028937, A028939.
Sequence in context: A024067 A192778 * A157101 A091001 A078811 A093399
Adjacent sequences: A051135 A051136 A051137 * A051139 A051140 A051141
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KEYWORD
| sign,easy,nice,changed
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AUTHOR
| Michael Somos, Oct 12 1999
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