login
This site is supported by donations to The OEIS Foundation.

 

Logo

Please make a donation to keep the OEIS running. We are now in our 55th year. In the past year we added 12000 new sequences and reached 8000 citations (which often say "discovered thanks to the OEIS"). We need to raise money to hire someone to manage submissions, which would reduce the load on our editors and speed up editing.
Other ways to donate

Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
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<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

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

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recent
The OEIS Community | Maintained by The OEIS Foundation Inc.

License Agreements, Terms of Use, Privacy Policy. .

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