

A135927


a(n) = a(n1)^2  2 with a(1) = 10.


2




OFFSET

1,1


COMMENTS

This is the LucasLehmer sequence with starting value u(1) = 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.


LINKS

Table of n, a(n) for n=1..7.
Raphael M. Robinson, Mersenne and Fermat Numbers, Proceedings of the American Mathematical Society, Vol. 5, No. 5. (October 1954), pp. 842846.


FORMULA

a(n) = 2*cosh(2^(n1)*log(5 + 2*sqrt(6))) = exp(2^(n1)*log(5 + 2*sqrt(6))) + exp(2^(n1)*log(5  2*sqrt(6))) = (5 + 2*sqrt(6))^(2^(n1)) + (5  2*sqrt(6))^(2^(n1)) = ceiling(exp(2^(n1)*log(5 + 2*sqrt(6)))) = ceiling((5 + 2*sqrt(6))^(2^(n1))).


EXAMPLE

2*cosh(2^3*log((5 + 2*sqrt(6))) = 92198402, so a(4) = 92198402.


MATHEMATICA

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


CROSSREFS

Cf. A000668, A000043, A003010, A095847, A001566, A135928.
Sequence in context: A254599 A217634 A007137 * A299952 A278672 A129542
Adjacent sequences: A135924 A135925 A135926 * A135928 A135929 A135930


KEYWORD

easy,nonn


AUTHOR

Ant King, Dec 07 2007


STATUS

approved



