A002248 Number of points on y^2 + xy = x^3 + x^2 + x over GF(2^n).

2, 8, 14, 16, 22, 56, 142, 288, 518, 968, 1982, 4144, 8374, 16472, 32494, 65088, 131174, 263144, 525086, 1047376, 2094358, 4193912, 8393806, 16783200, 33550022, 67092488, 134210174, 268460656, 536911222

Offset: 1

Author

N. J. A. Sloane, Iwan Duursma

Keywords

• nonn
• easy

Comments

This is a divisibility sequence; that is, if n divides m, then a(n) divides a(m). The point at infinity is counted also. - T. D. Noe, Mar 12 2009

Links

• Vincenzo Librandi, Table of n, a(n) for n = 1..1000
• Hugh Williams, R. K. Guy, Some fourth-order linear divisibility sequences, Intl. J. Number Theory vol. 7 (5) (2011) 1255-1277
• Index to divisibility sequences
• Index entries for linear recurrences with constant coefficients, signature (4,-7,8,-4).

Programs

Magma

I:=[2, 8, 14, 16]; [n le 4 select I[n] else 4*Self(n-1)-7*Self(n-2)+8*Self(n-3)-4*Self(n-4): n in [1..45]]; // Vincenzo Librandi, Jun 18 2012

Mathematica

Needs["FiniteFields`"]; Table[cnt=1; (* 1 point at infinity *) f=Table[GF[2,n][IntegerDigits[i,2,n]], {i,0,2^n-1}]; Do[If[y^2+x*y-x^3-x^2-x==0, cnt++ ], {x,f}, {y,f}]; cnt, {n,6}] (* T. D. Noe, Mar 12 2009 *)
LinearRecurrence[{4,-7,8,-4},{2,8,14,16},30] (* Vincenzo Librandi, Jun 18 2012 *)

PARI

a(n)=([0,1,0,0; 0,0,1,0; 0,0,0,1; -4,8,-7,4]^(n-1)*[2;8;14;16])[1,1] \\ Charles R Greathouse IV, Jun 23 2020

Formula

a(n) = 2^n + 1 - b(n); b(n) = b(n-1) - 2*b(n-2), b(1)=1, b(2)=-3; b(n) = A002249(n).

G.f.: -2*x*(-1+2*x^2) / ( (x-1)*(2*x-1)*(2*x^2 - x + 1) ).

a(n) = 4*a(n-1) - 7*a(n-2) + 8*a(n-3) - 4*a(n-4). - Vincenzo Librandi, Jun 18 2012