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A135927
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a(n) = a(n-1)^2 - 2 with a1=10.
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1
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OFFSET
| 1,1
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COMMENTS
| 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.
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REFERENCES
| Robinson, Raphael M.; Mersenne and Fermat Numbers, Proceedings of the American Mathematical Society, Vol. 5, No. 5. (October 1954), pp. 842-846.
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FORMULA
| 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))].
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EXAMPLE
| 2Cosh[2^3 Log[(5+2Sqrt(6))]=92198402, so a(4)=92198402
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MATHEMATICA
| a[1]=10; a[n_]:=a[n]=a[n-1]^2-2; a[ # ]&/@Range[7]
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CROSSREFS
| Cf. A000668, A000043, A003010, A095847, A001566, A135928.
Sequence in context: A163446 A190869 A007137 * A129542 A171315 A081109
Adjacent sequences: A135924 A135925 A135926 * A135928 A135929 A135930
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KEYWORD
| easy,nonn
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AUTHOR
| Ant King (mathstutoring(AT)ntlworld.com), Dec 07 2007
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