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A244373 a(n) = A242107(n+1) * A242107(n-1) * (1 + mod(n,2)).

%I A244373 #31 Sep 08 2022 08:46:08
%S A244373 1,0,1,2,-1,4,3,4,15,-14,65,224,-143,1824,1441,12882,50959,-151420,
%T A244373 898979,5337220,20799,188372002,-733599,6648401344,39471457217,
%U A244373 -234341035456,2785299158305,24790831385826,98497628929855,4377139749257604,-12158771603059997
%N A244373 a(n) = A242107(n+1) * A242107(n-1) * (1 + mod(n,2)).
%H A244373 G. C. Greubel, <a href="/A244373/b244373.txt">Table of n, a(n) for n = 0..232</a>
%F A244373 Given elliptic curve "58a1" : y^2 + x * y = x^3 - x^2 - x + 1, then the n th multiple of point [0, 1] is [a(n) / A242107(n)^2, A242107(n+2)^2 * A242107(n-4) / A242107(n)^3].
%F A244373 a(n) = a(-n) for all n in Z.
%F A244373 a(n+1) * A242107(n+4) = a(n+3) * A242107(n) for all n in Z.
%F A244373 0 = a(n)*a(n+7) + a(n+1)*a(n+6) - 2*a(n+2)*a(n+5) - 2*a(n+3)*a(n+4) for all n in Z.
%F A244373 0 = 2*a(n)*a(n+6) - a(n+1)*a(n+5) + 2*a(n+2)*a(n+4) - a(n+3)*a(n+3) for all even n in Z.
%F A244373 0 = a(n)*a(n+6) - 2*a(n+1)*a(n+5)  + a(n+2)*a(n+4) - 2*a(n+3)*a(n+3) for all odd n in Z.
%t A244373 Join[{1, 0}, RecurrenceTable[{a[n] == (-a[n-6]*a[n-1] + 2*a[n-2]*a[n-5] + 2*a[n-3]*a[n-4])/a[n-7], a[2] == 1, a[3] == 2, a[4] == -1, a[5] == 4, a[6] == 3, a[7] == 4, a[8] == 15}, a, {n, 2, 50}]] (* _G. C. Greubel_, Aug 05 2018 *)
%o A244373 (PARI) {a(n) = if( n==0, 1, n=abs(n); numerator( ellmul( ellinit([1, -1, 0, -1, 1]), [0, 1], n)[1]))};
%o A244373 (Magma) I:=[1,2,-1,4,3,4,15]; [n le 7 select I[n] else (-Self(n-6)*Self(n -1) + 2*Self(n-2)*Self(n-5) + 2*Self(n-3)*Self(n-4))/Self(n-7): n in [1..30]]; // _G. C. Greubel_, Aug 05 2018
%Y A244373 Cf. A242107.
%K A244373 sign
%O A244373 0,4
%A A244373 _Michael Somos_, Aug 22 2014