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A077496
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Decimal expansion of lim_{n -> infinity} A001699(n)^(1/2^n).
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4
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1, 5, 0, 2, 8, 3, 6, 8, 0, 1, 0, 4, 9, 7, 5, 6, 4, 9, 9, 7, 5, 2, 9, 3, 6, 4, 2, 3, 7, 3, 2, 1, 6, 9, 4, 0, 8, 7, 3, 8, 8, 7, 1, 7, 4, 3, 9, 6, 3, 5, 7, 9, 3, 0, 6, 9, 9, 0, 6, 7, 1, 4, 2, 4, 3, 0, 8, 4, 7, 1, 9, 7, 8, 7, 1, 7, 5, 7, 6, 6, 0, 1, 9, 4, 5, 6, 6, 3, 3, 3, 9, 1, 7, 8, 6, 3, 0, 6, 1, 9, 8, 7, 2, 3, 7
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OFFSET
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1,2
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REFERENCES
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Steven R. Finch, Mathematical Constants, Cambridge University Press, 2003, Section 6.10 Quadratic recurrence constants, p. 443.
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LINKS
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FORMULA
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Equals exp(Sum_{k>=1} log(1+1/A003095(k)^2)/2^k) (Aho and Sloane, 1973). - Amiram Eldar, Feb 02 2022
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EXAMPLE
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1.5028368010497564997529364237321694087388717439635793069906714243...
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MATHEMATICA
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digits = 105; Clear[b, beta]; b[0] = 1; b[n_] := b[n] = b[n-1]^2 + 1; b[10]; beta[n_] := beta[n] = b[n]^(2^(-n)); beta[5]; beta[n = 6]; While[ RealDigits[beta[n], 10, digits+5] != RealDigits[beta[n-1], 10, digits+5], Print["n = ", n]; n = n+1]; RealDigits[beta[n], 10, digits] // First (* Jean-François Alcover, Jun 18 2014 *)
(* Second program *)
S[n_]:= S[n]= If[n==1, Log[2]/2, S[n-1] + Log[1 + 1/A003095[n]^2]/2^n];
RealDigits[Exp[S[13]], 10, 120][[1]] (* G. C. Greubel, Nov 29 2022 *)
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PROG
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(Magma)
if n eq 0 then return 0;
end function;
function S(n)
if n eq 1 then return Log(2)/2;
else return S(n-1) + Log(1 + 1/A003095(n)^2)/2^n;
end if; return S;
end function;
SetDefaultRealField(RealField(120)); Exp(S(12)); // G. C. Greubel, Nov 29 2022
(SageMath)
@CachedFunction
@CachedFunction
def S(n): return log(2)/2 if (n==1) else S(n-1) + log(1 + 1/(A003095(n))^2)/2^n
numerical_approx( exp(S(12)), digits=120) # G. C. Greubel, Nov 29 2022
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CROSSREFS
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KEYWORD
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AUTHOR
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STATUS
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approved
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