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A135927 a(n) = a(n-1)^2 - 2 with a(1) = 10. 2
10, 98, 9602, 92198402, 8500545331353602, 72259270930397519221389558374402, 5221402235392591963136699520829303150191924374488750728808857602 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,1

COMMENTS

This is the Lucas-Lehmer 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. 842-846.

FORMULA

a(n) = 2*cosh(2^(n-1)*log(5 + 2*sqrt(6))) = exp(2^(n-1)*log(5 + 2*sqrt(6))) + exp(2^(n-1)*log(5 - 2*sqrt(6))) = (5 + 2*sqrt(6))^(2^(n-1)) + (5 - 2*sqrt(6))^(2^(n-1)) = ceiling(exp(2^(n-1)*log(5 + 2*sqrt(6)))) = ceiling((5 + 2*sqrt(6))^(2^(n-1))).

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

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Last modified February 20 12:34 EST 2020. Contains 332076 sequences. (Running on oeis4.)