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%I #40 Aug 23 2023 08:35:30
%S 0,4,8,4,16,44,56,116,288,508,968,2116,4144,8012,16472,33044,65088,
%T 130972,263144,523492,1047376,2099948,4193912,8383412,16783200,
%U 33558844,67092488,134225284,268460656,536830604,1073731736,2147574356,4294896768,8589823708
%N Number of points of a Koblitz curve E: y^2 + x*y = x^3 + a*x^2 + 1 over a field with 2^n elements.
%C A general result of Bilu, Hong, & Luca proves that, for n > e^e^10^10, a(n) has a prime divisor p > n*exp(0.0001*log n/log log n). - _Charles R Greathouse IV_, Feb 14 2022
%H G. C. Greubel, <a href="/A304293/b304293.txt">Table of n, a(n) for n = 0..1000</a>
%H Yuri Bilu, Haojie Hong, and Florian Luca, <a href="https://arxiv.org/abs/2112.07046">Big prime factors in orders of elliptic curves over finite fields</a>, arXiv:2112.07046 [math.NT], 2021.
%H J. H. Silverman, <a href="https://www.math.brown.edu/~jhs/Presentations/WyomingEllipticCurve.pdf">An Introduction to the Theory of Elliptic Curves</a>. See page 48.
%H <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (2,-1,4,-4).
%F G.f.: (4*x - 8*x^3) / (1 - 2*x + x^2 - 4*x^3 + 4*x^4).
%F a(n) = 2^n + 1 - ((-1 + 7 i)/2)^n - ((-1 - 7 i)/2)^n.
%F a(n) = a(-n) * 2^n for all n in Z.
%e G.f. = 4*x + 8*x^2 + 4*x^3 + 16*x^4 + 44*x^5 + 56*x^6 + 116*x^7 + ...
%t a[ n_] := Simplify[ 2^n + 1 - ((-1 + Sqrt[-7]) / 2)^n - ((-1 - Sqrt[-7]) / 2)^n];
%t CoefficientList[Series[(4*x-8*x^3)/(1-2*x+x^2-4*x^3+4*x^4), {x, 0, 50}], x] (* _G. C. Greubel_, Jul 28 2018 *)
%o (PARI) {a(n) = my(w=-quadgen(-7)); simplify(2^n + 1 - w^n - (-1-w)^n)};
%o (PARI) x='x+O('x^30); concat([0], Vec((4*x-8*x^3)/(1-2*x+x^2-4*x^3+ 4*x^4))) \\ _G. C. Greubel_, Jul 28 2018
%o (Magma) m:=30; R<x>:=PowerSeriesRing(Integers(), m); Coefficients(R!((4*x -8*x^3)/(1-2*x+x^2-4*x^3+4*x^4))); // _G. C. Greubel_, Jul 28 2018
%K nonn,easy
%O 0,2
%A _Michael Somos_, Jun 06 2018
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