1,1

This is the Lucas-Lehmer sequence with starting value u1=10 and the position of the zeros when it is reduced mod(2^p-1) also gives the position of the Mersenne primes. As we have started with n=1, these will occupy the (p-1)th positions in the sequence. For example, the first 12 terms mod(2^13-1) are 10, 98, 1411, 506, 2113, 672, 1077, 4996, 2037, 4721, 128, 0 and hence 8191 is a Mersenne prime. The radicals in the above closed forms are the solutions to x^2-10x+1=0.

Robinson, Raphael M.; Mersenne and Fermat Numbers, Proceedings of the American Mathematical Society, Vol. 5, No. 5. (October 1954), pp. 842-846.

a(n)=2 cosh[2^(n-1) Log(5+2Sqrt(6))]=Exp[2^(n-1) Log(5+2Sqrt(6))]+ Exp[2^(n-1) Log(5-2Sqrt(6))]= (5+2Sqrt(6))^(2^(n-1))+ (5-2Sqrt(6))^(2^(n-1))=Ceiling[Exp[2^(n-1) Log(5+2Sqrt(6))]]=Ceiling[(5+2Sqrt(6))^(2^(n-1))].

2Cosh[2^3 Log[(5+2Sqrt(6))]=92198402, so a(4)=92198402

a[1]=10; a[n_]:=a[n]=a[n-1]^2-2; a[ # ]&/@Range[7]

Cf. A000668, A000043, A003010, A095847, A001566, A135928.

Sequence in context: A125445 A163446 A007137 this_sequence A129542 A081109 A004189

Adjacent sequences: A135924 A135925 A135926 this_sequence A135928 A135929 A135930

easy,nonn

Ant King (mathstutoring(AT)ntlworld.com), Dec 07 2007