A267437 A linear recurrence related to the elliptic curves y^2 = x^3 -35*a^2*x - 98*a^3 with a = -1, -5, -6, -17, or -111.

11, 23, 67, 151, 275, 487, 963, 2039, 4211, 8327, 16291, 32407, 65363, 131623, 263043, 524087, 1046579, 2095559, 4196707, 8394199, 16778003, 33544039, 67096899, 134226551, 268468211, 536886023, 1073691427, 2147403031, 4294987475, 8590116007, 17180010243

Offset

2,1

Comments

Abatzoglou, Silverberg, Sutherland, & Wong give a quasi-quadratic algorithm for finding primes in this sequence, which relies on a correspondence between the Frobenius endomorphism of one of the five elliptic curves given above and complex multiplication in Z[(1 + sqrt(-7))/2].

Links

Charles R Greathouse IV, Table of n, a(n) for n = 2..3319

Alexander Abatzoglou, Alice Silverberg, Andrew V. Sutherland, and Angela Wong, Deterministic elliptic curve primality proving for a special sequence of numbers, Tenth Algorithmic Number Theory Symposium (ANTS X, 2012), pp. 1-20.

Alexander Abatzoglou, Alice Silverberg, Andrew V. Sutherland, Angela Wong, Deterministic elliptic curve primality proving for a special sequence of numbers, arXiv:1202.3695 [math.NT], 2012.

Index entries for linear recurrences with constant coefficients, signature (4,-7,8,-4).

Formula

a(n) = 4*a(n-1) - 7*a(n-2) + 8*a(n-3) - 4*a(n-4).

a(n) ~ 4*2^n.

G.f.: x^2*(11 - 21*x + 52*x^2 - 44*x^3)/((1 - x)*(1 - 2*x)*(1 - x + 2*x^2)). - Bruno Berselli, Jan 24 2016

a(n) = 1 + 2^(2+n) + 2*(1/2-(i*sqrt(7))/2)^n + 2*(1/2+(i*sqrt(7))/2)^n where i=sqrt(-1). - Colin Barker, Jul 02 2017

Mathematica

RecurrenceTable[{a[n] == 4 a[n - 1] - 7 a[n - 2] + 8 a[n - 3] - 4 a[n - 4], a[2] == 11, a[3] == 23, a[4] == 67, a[5] == 151}, a, {n, 2, 30}] (* Michael De Vlieger, Jan 24 2016 *)
LinearRecurrence[{4, -7, 8, -4}, {11, 23, 67, 151}, 40] (* Vincenzo Librandi, Jan 27 2016 *)

Prog

(PARI) a(n)=([0, 1, 0, 0; 0, 0, 1, 0; 0, 0, 0, 1; -4, 8, -7, 4]^n*[9; 11; 11; 23])[1, 1]
(PARI) first(n)=if(n<5, return(first(5)[1..n-1])); my(v=vector(n-1)); v[1]=11; v[2]=23; v[3]=67; v[4]=151; for(k=5, #v, v[k]=4*v[k-1]-7*v[k-2]+8*v[k-3]-4*v[k-4]); v
(Magma) I:=[11, 23, 67, 151]; [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..31]]; // Vincenzo Librandi, Jan 27 2016
(PARI) i=I; vector(50, n, n++; round(1 + 2^(2+n) + 2*(1/2-(i*sqrt(7))/2)^n + 2*(1/2+(i*sqrt(7))/2)^n)) \\ Colin Barker, Jul 02 2017

Crossrefs

Cf. A267438, A267439.

Sequence in context: A081510 A068844 A139905 * A267438 A102273 A195463

Adjacent sequences: A267434 A267435 A267436 * A267438 A267439 A267440

Keyword

nonn,easy

Author

Charles R Greathouse IV, Jan 15 2016

Status

approved