A242232 a(n) = 2*a(n-1)^2 - 1, a(0)=6.

6, 71, 10081, 203253121, 82623662392481281, 13653339174293451118767199870801921, 372827341216592355174245573447441869623455324379507680549087234580481

Offset: 0

Author

Vaclav Kotesovec, May 08 2014

Keywords

• nonn

Comments

In general, for a(0)=p is a(n) = cosh(2^n*arccosh(p)) = (1/2)*(p+sqrt(p^2-1))^(2^n) + (1/2)*(p+sqrt(p^2-1))^(-2^n).

Crossrefs

Cf. A002812, A001601, A005828, A084765.

Programs

Mathematica

RecurrenceTable[{a[n+1]==2*a[n]^2-1,a[0]==6},a,{n,0,10}]
NestList[2#^2-1&,6,10] (* Harvey P. Dale, Jun 12 2025 *)

Formula

a(n) = (1/2)*(6+sqrt(35))^(2^n) + (1/2)*(6+sqrt(35))^(-2^n).

a(n) = A023038(2^n).

a(n) = T(2^n,6), where T(n,x) denotes the n-th Chebyshev polynomial of the first kind. - Peter Bala, Mar 30 2022