A346625 - OEIS

%I #29 Sep 08 2022 08:46:26

%S 8,62,3842,14760962,217885999165442,47474308632322991920487055362,

%T 2253809980117057347661794063813616885861274573005652951042

%N a(0) = 8; for n > 0, a(n) = a(n-1)^2 - 2.

%C For n >= 2, the Fermat number F(n) = 2^(2^n) + 1 is prime if and only if F(n) divides a(2^n-2).

%D Kusta Inkeri, Tests for primality, Ann. Acad. Sci. Fenn., A I No. 279 (1960), pp. 1-19.

%D M. Krizek, F. Luca, L. Somer, 17 Lectures on Fermat Numbers: From Number Theory to Geometry, CMS Books in Mathematics, vol. 9, Springer-Verlag, New York, 2001, p. 46.

%H <a href="/index/Aa#AHSL">Index entries for sequences of form a(n+1)=a(n)^2 + ...</a>

%F For n >= 2, a(n) = 2 + Sum_{k=1..2^(n-1)} (-1)^k*64^k*2^(n-1)*binomial(k + 2^(n-1) - 1, 2*k - 1)/k.

%F a(n) = ceiling(c^(2^n)) where c = 4 + sqrt(15) is the largest root of x^2 - 8*x + 1 = 0.

%F a(n) = (4 + sqrt(15))^(2^n) + (4 - sqrt(15))^(2^n).

%F a(n) = 2*T(2^n,4), where T(n,x) denotes the n-th Chebyshev polynomial of the first kind.

%F sqrt(5/3) = Product_{n >= 0} (1 + 2/a(n)).

%F 2*sqrt(5/3)/3 = Product_{n >= 0} (1 - 1/a(n)).

%F a(n) = 2*A005828(n).

%F a(n) = A086903(2^n) = A220337(3*(n+1)).

%t NestList[#^2 - 2 &, 8, 6]

%o (Magma) [8] cat [n eq 1 select 62 else Self(n-1)^2-2: n in [1..6]];

%o (PARI) {a(n)=if(n<1, 8*(n==0), a(n-1)^2-2)};

%Y Cf. A000215, A086903, A304177.

%K nonn,easy

%O 0,1

%A _Arkadiusz Wesolowski_, Jul 26 2021