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A084764
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a(n) = 2*a(n-1)^2 - 1, a(0)=1, a(1)=4.
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4
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OFFSET
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0,2
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COMMENTS
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Product_{k=1..n} (1 + 1/a(k)) converges to sqrt(5/3).
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LINKS
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FORMULA
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With x=4+sqrt(15), y=4-sqrt(15): a(n+1) = (x^(2^n) + y^(2^n))/2.
a(n) = A001091(2^(n-1)) with a(0) = 1; i.e. a(n) = ChebyshevT(2^(n-1), 4) with a(0) = 1. - G. C. Greubel, May 16 2023
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MATHEMATICA
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a[n_]:= a[n]= If[n<2, 4^n, 2 a[n-1]^2 -1]; Table[a[n], {n, 0, 10}]
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PROG
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(Magma) [n le 2 select 4^(n-1) else 2*Self(n-1)^2 - 1: n in [1..10]]; // G. C. Greubel, May 16 2023
(SageMath)
def A084764(n): return 1 if n==0 else chebyshev_T(2^(n-1), 4)
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CROSSREFS
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KEYWORD
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easy,nonn
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AUTHOR
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Mario Catalani (mario.catalani(AT)unito.it), Jun 04 2003
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STATUS
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approved
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