cp's OEIS Frontend

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

%I A002248 #30 Sep 08 2022 08:44:30
%S A002248 2,8,14,16,22,56,142,288,518,968,1982,4144,8374,16472,32494,65088,
%T A002248 131174,263144,525086,1047376,2094358,4193912,8393806,16783200,
%U A002248 33550022,67092488,134210174,268460656,536911222
%N A002248 Number of points on y^2 + xy = x^3 + x^2 + x over GF(2^n).
%C A002248 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
%H A002248 Vincenzo Librandi, <a href="/A002248/b002248.txt">Table of n, a(n) for n = 1..1000</a>
%H A002248 Hugh Williams, R. K. Guy, <a href="http://dx.doi.org/10.1142/S1793042111004587">Some fourth-order linear divisibility sequences</a>, Intl. J. Number Theory vol. 7 (5) (2011) 1255-1277
%H A002248 <a href="/index/Di#divseq">Index to divisibility sequences</a>
%H A002248 <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (4,-7,8,-4).
%F A002248 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).
%F A002248 G.f.: -2*x*(-1+2*x^2) / ( (x-1)*(2*x-1)*(2*x^2 - x + 1) ).
%F A002248 a(n) = 4*a(n-1) - 7*a(n-2) + 8*a(n-3) - 4*a(n-4). - _Vincenzo Librandi_, Jun 18 2012
%t A002248 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 *)
%t A002248 LinearRecurrence[{4,-7,8,-4},{2,8,14,16},30] (* _Vincenzo Librandi_, Jun 18 2012 *)
%o A002248 (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
%o A002248 (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
%K A002248 nonn,easy
%O A002248 1,1
%A A002248 _N. J. A. Sloane_, _Iwan Duursma_