A259419 A Somos-4 like sequence connected with the elliptic curve y^2 + y = x^3 - 3x + 4.

1, 2, 1, -3, -7, -17, 2, 101, 247, 571, -1669, -13766, -43101, -205897, 1640929, 8217293, 101727662, 173114917, -5439590147, -70987557871, -993401657957, 2105332714614, 208894282701569, 3110590234593151, 37334338699443371, -891470356091782993, -33292234868859857114

Offset

0,2

Comments

All the terms of the sequence are integers. Moreover, a(n)^2 is the denominator of the x-coordinate of (2n+3)P, where P = (4,7) is the point on the elliptic curve E: y^2 + y = x^3 - 3x + 4.

Links

Seiichi Manyama, Table of n, a(n) for n = 0..195

Alexi Block Gorman, Tyler Genao, Heesu Hwang, Noam Kantor, Sarah Parsons, and Jeremy Rouse, The density of primes dividing a particular non-linear recurrence sequence, arXiv:1508.02464 [math.NT], 2015.

Clark Kimberling, Strong divisibility sequences and some conjectures, Fib. Quart., 17 (1979), 13-17.

PUMaC, PUMac 2015 Power Round. See page 21, denoted by d_n.

Formula

a(n) = (a(n-1)*a(n-3) - a(n-2)^2)/a(n-4) if n is not 2 mod 3, and a(n) = (a(n-1)*a(n-3) - 3*a(n-2)^2)/a(n-4) if n is 2 mod 3.

a(n) = - a(-3-n) for all n in Z. - Michael Somos, Aug 13 2015

a(n)*a(n+7) = -1*a(n+1)*a(n+6) +5*a(n+3)*a(n+4) for all n in Z. - Michael Somos, Aug 13 2015

a(n)*a(n+8) = -4*a(n+2)*a(n+6) +5*a(n+3)*a(n+5) for all n in Z. - Michael Somos, Aug 13 2015

Let t(n) be a strong elliptic divisibility sequence as given in [Kimberling, p. 16] where x = 5^(1/4), y = 3^(1/3), z = 1. Then a(n) = t(2*n + 3) / if( 3|n, y, 1). - Michael Somos, Aug 13 2015

Example

If P = (4,7), then (2*4+3)*P = (-104/49, 181/343). The denominator of the x-coordinate is 49 = a(4)^2.

Mathematica

a[ n_] := Module[ {v, m, s = 1}, m = If[ n < -1, s = -1; -3 - n, n] + 5; v = Join[{-2, -1, -1, 1, 1, 2, 1}, Table[0, {m - 7}]]; Do[ v[[k]] = (5 v[[k - 3]] v[[k - 4]] - v[[k - 1]] v[[k - 6]]) / v[[k - 7]], {k, 8, m}]; s v[[m]]]; (* Michael Somos, Aug 13 2015 *)

Prog

(PARI) a = vector(99); a[1]=2; a[2]=1; a[3] = -3; a[4] = -7; for(n=5, #a, if(Mod(n, 3)==Mod(2, 3), a[n]=(a[n-1]*a[n-3]-3*a[n-2]^2)/a[n-4], a[n]=(a[n-1]*a[n-3]-a[n-2]^2)/a[n-4])); a
(PARI) {a(n) = my(v, s=1); if( n<-1, n = -3-n; s = -1); n += 5; v = concat( [-2, -1, -1, 1, 1, 2, 1], vector( max(0, n-7))); for(k=8, n, v[k] = (5 * v[k-3] * v[k-4] - v[k-1] * v[k-6]) / v[k-7]); s * v[n]}; /* Michael Somos, Aug 13 2015 */

Crossrefs

Cf. A006720.

Sequence in context: A137766 A294620 A010758 * A019224 A298096 A053190

Adjacent sequences: A259416 A259417 A259418 * A259420 A259421 A259422

Keyword

easy,sign

Author

Jeremy Rouse, Jun 26 2015

Status

approved