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A077496 Decimal expansion of lim_{n -> infinity} A001699(n)^(1/2^n).

%I A077496 #34 Feb 16 2025 08:32:48
%S A077496 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,
%T A077496 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,
%U A077496 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
%N A077496 Decimal expansion of lim_{n -> infinity} A001699(n)^(1/2^n).
%D A077496 Steven R. Finch, Mathematical Constants, Cambridge University Press, 2003, Section 6.10 Quadratic recurrence constants, p. 443.
%H A077496 G. C. Greubel, <a href="/A077496/b077496.txt">Table of n, a(n) for n = 1..10000</a>
%H A077496 A. V. Aho and N. J. A. Sloane, <a href="https://web.archive.org/web/2024*/https://www.fq.math.ca/Scanned/11-4/aho-a.pdf">Some doubly exponential sequences</a>, Fibonacci Quarterly, Vol. 11, No. 4 (1973), pp. 429-437.
%H A077496 A. V. Aho and N. J. A. Sloane, <a href="/A000058/a000058.pdf">Some doubly exponential sequences</a>, Fibonacci Quarterly, Vol. 11, No. 4 (1973), pp. 429-437 (original plus references that F.Q. forgot to include - see last page!)
%H A077496 Anna de Mier and Marc Noy, <a href="https://doi.org/10.1016/j.endm.2009.07.081">On the maximum number of cycles in outerplanar and series-parallel graphs</a>, El. Notes Discr. Math., Vol. 34 (2009), pp. 489-493, Proposition 2.2.
%H A077496 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/QuadraticRecurrenceEquation.html">Quadratic Recurrence Equation</a>.
%F A077496 Equals A076949^2. - _Vaclav Kotesovec_, Dec 17 2014
%F A077496 Equals exp(Sum_{k>=1} log(1+1/A003095(k)^2)/2^k) (Aho and Sloane, 1973). - _Amiram Eldar_, Feb 02 2022
%e A077496 1.5028368010497564997529364237321694087388717439635793069906714243...
%t A077496 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 *)
%t A077496 (* Second program *)
%t A077496 A003095[n_]:= A003095[n]= If[n==0, 0, 1 + A003095[n-1]^2];
%t A077496 S[n_]:= S[n]= If[n==1, Log[2]/2, S[n-1] + Log[1 + 1/A003095[n]^2]/2^n];
%t A077496 RealDigits[Exp[S[13]], 10, 120][[1]] (* _G. C. Greubel_, Nov 29 2022 *)
%o A077496 (Magma)
%o A077496 function A003095(n)
%o A077496   if n eq 0 then return 0;
%o A077496   else return 1 + A003095(n-1)^2;
%o A077496   end if; return A003095;
%o A077496 end function;
%o A077496 function S(n)
%o A077496   if n eq 1 then return Log(2)/2;
%o A077496   else return S(n-1) + Log(1 + 1/A003095(n)^2)/2^n;
%o A077496   end if; return S;
%o A077496 end function;
%o A077496 SetDefaultRealField(RealField(120)); Exp(S(12)); // _G. C. Greubel_, Nov 29 2022
%o A077496 (SageMath)
%o A077496 @CachedFunction
%o A077496 def A003095(n): return 0 if (n==0) else 1 + A003095(n-1)^2
%o A077496 @CachedFunction
%o A077496 def S(n): return log(2)/2 if (n==1) else S(n-1) + log(1 + 1/(A003095(n))^2)/2^n
%o A077496 numerical_approx( exp(S(12)), digits=120) # _G. C. Greubel_, Nov 29 2022
%Y A077496 Cf. A001699, A003095, A076949.
%K A077496 cons,nonn
%O A077496 1,2
%A A077496 _Benoit Cloitre_, Dec 01 2002