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A002248
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Number of points on y^2 + xy = x^3 + x^2 + x over GF(2^n).
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2
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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
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OFFSET
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1,1
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COMMENTS
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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
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LINKS
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FORMULA
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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) ).
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MATHEMATICA
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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 *)
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PROG
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(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
(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
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CROSSREFS
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KEYWORD
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nonn,easy
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AUTHOR
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STATUS
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approved
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