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A135927
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a(n) = a(n-1)^2 - 2 with a(1) = 10.
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6
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OFFSET
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1,1
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COMMENTS
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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.
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LINKS
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Gabriel Klambauer, Summation of Series, Amer. Math. Monthly, Vol. 87, No. 2 (Feb., 1980), pp. 128-130.
Raphael M. Robinson, Mersenne and Fermat Numbers, Proceedings of the American Mathematical Society, Vol. 5, No. 5. (October 1954), pp. 842-846.
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FORMULA
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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))).
Product_{n >= 1} (1 + 2/a(n)) = (1/2)*sqrt(6); Product_{n >= 1} (1 - 1/a(n)) = (4/11)*sqrt(6).
Engel expansion of 5 - sqrt(24) = 1/a(1) + 1/(a(1)*a(2)) + 1/(a(1)*a(2)*a(3)) + .... See Klambauer, p. 130. (End)
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EXAMPLE
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2*cosh(2^3*log((5 + 2*sqrt(6))) = 92198402, so a(4) = 92198402.
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MATHEMATICA
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a[1] = 10; a[n_] := a[n] = a[n - 1]^2 - 2; a[#]&/@Range[7]
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PROG
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(Python)
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CROSSREFS
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KEYWORD
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nonn,easy
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AUTHOR
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STATUS
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approved
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