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%I #34 Sep 08 2022 08:46:00
%S 22,190,2878,3862,111382,117118,3864190,3897622,131738902,131933758,
%T 4477986238,4479121942,152135692822,152142312190,5168228240638,
%U 5168266821142,175568164615702,175568389479358,5964152516784190,5964153827385622,202605635754466582
%N Successive values x of solutions Mordell's elliptic curve x^3-y^2 = d contained points {x,y} with quadratic extension sqrt(2) over rationals.
%C This sequence is equivalent of A200216, but A200216 was for quadratic field with extension sqrt(5).
%C Coefficient r=sqrt(x)/d tend to sqrt(2)/432 ~ 0.00327364 when x and d tend to infinity.
%C Starting from a(2)= 2878 all points are extremal (for definition see A200656).
%C (a(n)+10)/2 is perfect square of even number for each n.
%C All numbers in this sequence are of the form 2*(12*k+11).
%C For y values see A200937.
%C For d values see A200938.
%C When n is even d=A200938(n) are positive~, when n is odd d=A200938(n) are negative.
%H G. C. Greubel, <a href="/A200936/b200936.txt">Table of n, a(n) for n = 0..1000</a>
%F a(n) = (A200937(n)^2 + A200938(n))^(1/3).
%F a(n) = a(n-1)+ 40*a(n-2) - 40*a(n-3) - 206*a(n-4) + 206*a(n-5) + 40*a(n-6) - 40*a(n-7) - a(n-8) + a(n-9).
%F G.f.: 2*(11+84*z+904*z^2-2868*z^3+492*z^5-12*z^7+2266*z^4-440*z^6 +11*z^8)/((1-z)*(z^2+6*z+1)*(1-6*z+z^2)*(z^2+2*z-1)*(z^2-2*z-1)).
%F a(2*n + 1) - a(2*n) = 24 * A001333(2*n + 3), a(n) = a(-5-n) for all n in Z. - _Michael Somos_, Aug 23 2018
%e a(3)=2878=A200656(1) because 2878^3-154396^2=15336.
%e G.f. = 22 + 190*x + 2868*x^2 + 3862*x^3 + 111382*x^4 + 117118*x^5 + ... - _Michael Somos_, Aug 23 2018
%t aa = {22, 190, 2878, 3862, 111382, 117118, 3864190, 3897622, 131738902}; a1 = aa[[1]]; a2 = aa[[3]]; a3 = aa[[3]]; a4 = aa[[4]]; a5 = aa[[5]]; a6 = aa[[6]]; a7 = aa[[7]]; a8 = aa[[8]]; a9 = aa[[9]]; Do[an = a9 + 40*a8 - 40*a7 - 206*a6 + 206*a5 + 40*a4 - 40*a3 - a2 + a1; a1 = a2; a2 = a3; a3 = a4; a4 = a5; a5 = a6; a6 = a7; a7 = a8; a8 = a9; a9 = an; AppendTo[aa, an], {nn, 20}]; aa
%t CoefficientList[Series[-2*(11 + 84*z + 904*z^2 - 2868*z^3 + 492*z^5 - 12*z^7 + 2266*z^4 - 440*z^6 + 11*z^8)/((z - 1) (z^2 + 6*z + 1) (1 - 6*z + z^2) (z^2 + 2*z - 1) (z^2 - 2*z - 1)), {z, 0, 30}], z] (* _G. C. Greubel_, Jul 27 2018 *)
%t a[ n_] := With[{m = Max[-5 - n, n]}, SeriesCoefficient[ 2 (1 - 12 x - 40 x^2 + 396 x^3 - 1138 x^4 + 396 x^5 - 40 x^6 - 12 x^7 + x^8) / (x^2 (x - 1) (1 + 6 x + x^2) (1 - 6 x + x^2) (x^2 + 2 x - 1) (x^2 - 2 x - 1)), {x, 0, m}]]; (* _Michael Somos_, Aug 23 2018 *)
%t a[ n_] := With[ {m = If[ OddQ[n], -5 - n, n], r1 = 1 + Sqrt[2], r2 = 1 - Sqrt[2]}, Simplify[7 - 6 (6 r1 + r2) r1^m - 6 (r1 + 6 r2) r2^m + (169 r1 + 29 r2)/4 r1^(2 m) + (29 r1 + 169 r2)/4 r2^(2 m)]]; (* _Michael Somos_, Aug 25 2018 *)
%o (PARI) z='z+O('z^30); Vec(2*(11+84*z+904*z^2-2868*z^3+492*z^5 -12*z^7 +2266*z^4 -440*z^6 +11*z^8)/((1-z)*(z^2+6*z+1)*(1-6*z+z^2)*(z^2+2*z-1)*(z^2-2*z-1))) \\ _G. C. Greubel_, Jul 27 2018
%o (PARI) {a(n) = my(m = max(-5-n, n)); polcoeff( 2*(1 - 12*x - 40*x^2 + 396*x^3 - 1138*x^4 + 396*x^5 - 40*x^6 - 12*x^7 + x^8) / (x^2*(x - 1)*(1 + 6*x + x^2)*(1 - 6*x + x^2)*(x^2 + 2*x - 1)*(x^2 - 2*x - 1)) + x * O(x^m), m)}; /* _Michael Somos_, Aug 23 2018 */
%o (PARI) {a(n) = my(m = if(n%2, -5-n, n), r1 = 1 + quadgen(8), r2 = 1 - quadgen(8)); simplify(7 - 6*(6*r1 + r2) * r1^m - 6*(r1 + 6*r2) * r2^m + (169*r1 + 29*r2)/4 * r1^(2*m) + (29*r1 + 169*r2)/4 * r2^(2*m))}; /* _Michael Somos_, Aug 25 2018 */
%o (Magma) m:=30; R<z>:=PowerSeriesRing(Integers(), m); Coefficients(R!(2*(11+84*z+904*z^2-2868*z^3+492*z^5 -12*z^7 +2266*z^4 -440*z^6 +11*z^8)/((1-z)*(z^2+6*z+1)*(1-6*z+z^2)*(z^2+2*z-1)*(z^2-2*z-1)))); // _G. C. Greubel_, Jul 27 2018
%Y Cf. A001333, A200216, A200656.
%K nonn
%O 0,1
%A _Artur Jasinski_, Nov 25 2011
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