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A178418 A (-1, 2) Somos-4 sequence associated to the elliptic curve y^2 + 2*x*y + y = x^3 + x^2 + x. 1

%I #18 Sep 08 2022 08:45:54

%S 1,1,2,9,-1,164,-737,5895,-119558,-113489,-39697759,-800750664,

%T -25602199327,-2344309254991,14264330936074,-14182654502256615,

%U 1282764246790221919,-163799892405003723284,67851128001519788451263

%N A (-1, 2) Somos-4 sequence associated to the elliptic curve y^2 + 2*x*y + y = x^3 + x^2 + x.

%C Hankel transform of the sequence with g.f. 1/(1-x^2/(1-2x^2/(1-(9/4)x^2/(1+(2/81)x^2/(1-1476x^2/(1-.... where 0/1, 2/1, 9/4, -2/81, 1476/1,... are the x-coordinates of the multiples of (0, 0).

%H G. C. Greubel, <a href="/A178418/b178418.txt">Table of n, a(n) for n = 1..125</a> (offset adapted by _Georg Fischer_, Jan 31 2019)

%F a(n) = (-a(n-1)*a(n-3) + 2*a(n-2)^2)/a(n-4), n>4.

%F a(n) = -(-1)^n * a(-n), a(n+3)*a(n-2) = -a(n+2)*a(n-1) + 9*a(n+1)*a(n) for all n in Z. - _Michael Somos_, Sep 19 2018

%e G.f. = x + x^2 + 2*x^3 + 9*x^4 - x^5 + 164*x^6 - 737*x^7 + ... - _Michael Somos_, Sep 19 2018

%t RecurrenceTable[{a[1]==a[2]==1,a[3]==2,a[4]==9,a[n]==(-a[n-1]a[n-3]+ 2a[n-2]^2)/a[n-4]},a[n],{n,20}] (* _Harvey P. Dale_, Sep 20 2011 *)

%o (PARI) m=30; v=concat([1,1,2,9], vector(m-4)); for(n=5, m, v[n] = ( -v[n-1]*v[n-3] + 2*v[n-2]^2)/v[n-4]); v \\ _G. C. Greubel_, Sep 18 2018

%o (Magma) I:=[1,1,2,9]; [n le 4 select I[n] else (-Self(n-1)*Self(n-3) + 2*Self(n-2)^2)/Self(n-4): n in [1..30]]; // _G. C. Greubel_, Sep 18 2018

%K easy,sign

%O 1,3

%A _Paul Barry_, May 27 2010

%E Offset changed to 1 by _Michael Somos_, Sep 19 2018

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Last modified March 28 17:25 EDT 2024. Contains 371254 sequences. (Running on oeis4.)