A259419 - OEIS

%I #30 Jan 05 2025 19:51:40

%S 1,2,1,-3,-7,-17,2,101,247,571,-1669,-13766,-43101,-205897,1640929,

%T 8217293,101727662,173114917,-5439590147,-70987557871,-993401657957,

%U 2105332714614,208894282701569,3110590234593151,37334338699443371,-891470356091782993,-33292234868859857114

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

%C 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.

%H Seiichi Manyama, <a href="/A259419/b259419.txt">Table of n, a(n) for n = 0..195</a>

%H Alexi Block Gorman, Tyler Genao, Heesu Hwang, Noam Kantor, Sarah Parsons, and Jeremy Rouse, <a href="http://arxiv.org/abs/1508.02464">The density of primes dividing a particular non-linear recurrence sequence</a>, arXiv:1508.02464 [math.NT], 2015.

%H Clark Kimberling, <a href="https://web.archive.org/web/2024*/https://www.fq.math.ca/Scanned/17-1/kimberling1.pdf">Strong divisibility sequences and some conjectures</a>, Fib. Quart., 17 (1979), 13-17.

%H PUMaC, <a href="https://jason-shi-f9dm.squarespace.com/s/PUMaC2015_PowerwithSol.pdf">PUMac 2015 Power Round</a>. See page 21, denoted by d_n.

%F 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.

%F a(n) = - a(-3-n) for all n in Z. - _Michael Somos_, Aug 13 2015

%F 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

%F 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

%F 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

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

%t 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 *)

%o (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

%o (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 */

%Y Cf. A006720.

%K easy,sign

%O 0,2

%A _Jeremy Rouse_, Jun 26 2015