A378683 a(0) = 1, a(n+1) = 6*a(n)^3 - 3*a(n).

1, 3, 153, 21489003, 59538796254981950751153, 1266343134315970349117919634635229303292221774134557782012151266098003

Offset: 0

Author

Robert FERREOL, Dec 03 2024

Keywords

• nonn

Comments

If we define u(0) = 1 , u(n+1) = (u(n)/3)*(u(n)^2+9) / (u(n)^2 + 1), then u(n) = A238799(n) / a(n) ; this is Halley's method to calculate sqrt(3).

Links

• Wikipedia, Halley's method.

Crossrefs

Cf. A002530, A002605, A238799.

Programs

Maple

a:=1 : A:=NULL : for k to 5 do a:=6*a^3-3*a : A:=A,a od : A;

Mathematica

NestList[6#^3-3# &, 1, 5] (* Stefano Spezia, Dec 06 2024 *)

Formula

a(n) = ((1 + sqrt(3))^(3^n) - (1 - sqrt(3))^(3^n))/2^((3^n+1)/2) / sqrt(3).

a(n) = A002530(3^n).

a(n) = A002605(3^n)/2^((3^n+1)/2).