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 A304293 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. 1
 0, 4, 8, 4, 16, 44, 56, 116, 288, 508, 968, 2116, 4144, 8012, 16472, 33044, 65088, 130972, 263144, 523492, 1047376, 2099948, 4193912, 8383412, 16783200, 33558844, 67092488, 134225284, 268460656, 536830604, 1073731736, 2147574356, 4294896768, 8589823708 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,2 LINKS G. C. Greubel, Table of n, a(n) for n = 0..1000 J. H. Silverman, An Introduction to the Theory of Elliptic Curves. See page 48. Index entries for linear recurrences with constant coefficients, signature (2,-1,4,-4). FORMULA G.f.: (4*x - 8*x^3) / (1 - 2*x + x^2 - 4*x^3 + 4*x^4). a(n) = 2^n + 1 - ((-1 + 7 i)/2)^n - ((-1 - 7 i)/2)^n. a(n) = a(-n) * 2^n for all n in Z. EXAMPLE G.f. = 4*x + 8*x^2 + 4*x^3 + 16*x^4 + 44*x^5 + 56*x^6 + 116*x^7 + ... MATHEMATICA a[ n_] := Simplify[ 2^n + 1 - ((-1 + Sqrt[-7]) / 2)^n -  ((-1 - Sqrt[-7]) / 2)^n]; 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 *) PROG (PARI) {a(n) = my(w=-quadgen(-7)); simplify(2^n + 1 - w^n - (-1-w)^n)}; (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 (MAGMA) m:=30; R:=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 CROSSREFS Sequence in context: A028587 A087260 A019254 * A055374 A275876 A255293 Adjacent sequences:  A304290 A304291 A304292 * A304294 A304295 A304296 KEYWORD nonn,easy AUTHOR Michael Somos, Jun 06 2018 STATUS approved

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Last modified December 8 18:37 EST 2019. Contains 329865 sequences. (Running on oeis4.)