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A377310 Divisibility sequence associated with elliptic curve y^2 + y = x^3 - x^2 - 2x + 2 and point (1, 0).

%I A377310 #12 May 05 2025 18:01:37
%S A377310 0,1,1,1,-3,-4,-13,23,87,415,-152,-8063,-38727,-142471,2309453,
%T A377310 13609844,187790979,-1743980081,-25547499185,-575984295329,
%U A377310 1873521429456,217675476797921,5045023692031697,65853623974941521,-5934036772012185603,-157454833217800083092
%N A377310 Divisibility sequence associated with elliptic curve y^2 + y = x^3 - x^2 - 2x + 2 and point (1, 0).
%C A377310 Bisection of A210098 (even part). The other bisection is A277279.
%C A377310 The elliptic curve y^2 + y = x^3 - x^2 - 2x + 2 has LMFDB label 57.a1 (Cremona label 57a1).
%C A377310 This is a strong elliptic divisibility sequence t_n as given in [Kimberling, p. 16] where x = 1, y = 1, z = -3.
%H A377310 C. 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 A377310 LMFDB, <a href="https://www.lmfdb.org/EllipticCurve/Q/57a1">Elliptic Curve 57a1</a>
%F A377310 a(n) = A210098(2*n).
%F A377310 a(n) = -a(-n) for all n in Z.
%F A377310 0 = a(n)*a(n+4) - a(n+1)*a(n+3) + a(n+2)^2 for all n in Z.
%F A377310 0 = a(n)*a(n+5) - a(n+1)*a(n+4) - 3*a(n+2)*a(n+3) for all n in Z.
%F A377310 0 = a(n+1)^2*a(n+2)^2 - a(n)^2*a(n+3)^2 - a(n)*a(n+2)^3 - a(n+1)^3*a(n+3) - 2*a(n)*a(n+1)*a(n+2)*a(n+3) for all n in Z.
%e A377310 G.f. = x + x^2 + x^3 - 3*x^4 - 4*x^5 - 13*x^6 + 23*x^7 + 87*x^8 + 415*x^9 + ...
%p A377310 b:= proc(n) option remember; `if`(n<6, [0, 1$4, 2][n+1],
%p A377310       (b(n-1)*b(n-4) -b(n-2)*b(n-3)) / b(n-5))
%p A377310     end:
%p A377310 a:= n-> b(2*n):
%p A377310 seq(a(n), n=0..25);  # _Alois P. Heinz_, May 05 2025
%t A377310 a[ n_] := a[n] = Which[ n<0, -a[-n], n<5, {0, 1, 1, 1, -3}[[n+1]], True, (a[n-1]*a[n-3] - a[n-2]^2)/a[n-4]];
%o A377310 (PARI) {a(n) = my(v); if(n<0, -a(-n), n<5, [0, 1, 1, 1, -3][n+1], v = vector(n, i, if(i<5, a(i))); for(i=5, n, v[i] = (v[i-1]*v[i-3] - v[i-2]^2)/v[i-4]); v[n])};
%o A377310 (PARI) {a(n) = my(E = ellinit([0, -1, 1, -2, 2]), z); z = ellpointtoz(E, [1, 0]); -(-1)^n*round(ellsigma(E, n*z)/ellsigma(E, z)^(n^2))};
%o A377310 (PARI) {a(n) = my(E = ellinit([0, -1, 1, -2, 2])); sign(n) * subst( elldivpol( E, abs(n)), x, 1)};
%Y A377310 Cf. A210098, A277279.
%K A377310 sign
%O A377310 0,5
%A A377310 _Michael Somos_, Oct 23 2024
%E A377310 Duplicate term a(15)=2309453 removed by _Georg Fischer_, May 05 2025