cp's OEIS Frontend

A076949 Decimal expansion of c, the constant such that lim n -> infinity A003095(n)/c^(2^n) = 1.

%I A076949 #23 Nov 30 2022 08:26:12
%S A076949 1,2,2,5,9,0,2,4,4,3,5,2,8,7,4,8,5,3,8,6,2,7,9,4,7,4,9,5,9,1,3,0,0,8,
%T A076949 5,2,1,3,2,1,2,2,9,3,2,0,9,6,9,6,6,1,2,8,2,3,1,7,7,0,0,9,0,7,2,5,5,2,
%U A076949 3,3,9,9,7,5,2,6,5,7,3,8,0,2,1,6,7,9,1,5,4,5,2,0,4,9,9,2,8,4,2,9
%N A076949 Decimal expansion of c, the constant such that lim n -> infinity A003095(n)/c^(2^n) = 1.
%H A076949 G. C. Greubel, <a href="/A076949/b076949.txt">Table of n, a(n) for n = 1..10000</a>
%H A076949 Stephan Wagner, Volker Ziegler, <a href="https://arxiv.org/abs/2004.09353">Irrationality of growth constants associated with polynomial recursions</a>, arXiv:2004.09353 [math.NT], 2020.
%F A076949 Equals sqrt(A077496). - _Vaclav Kotesovec_, Dec 17 2014
%e A076949 1.2259024435287485386279474959130085213212293209696612823177009072552339975...
%t A076949 A003095[n_]:= A003095[n]= If[n==0, 0, 1 + A003095[n-1]^2];
%t A076949 S[n_]:= S[n]= If[n==1, Log[2]/2, S[n-1] + Log[1 + 1/A003095[n]^2]/2^n];
%t A076949 RealDigits[Exp[S[13]/2], 10, 120][[1]] (* _G. C. Greubel_, Nov 29 2022 *)
%o A076949 (Magma)
%o A076949 function A003095(n)
%o A076949   if n eq 0 then return 0;
%o A076949   else return 1 + A003095(n-1)^2;
%o A076949   end if; return A003095;
%o A076949 end function;
%o A076949 function S(n)
%o A076949   if n eq 1 then return Log(2)/2;
%o A076949   else return S(n-1) + Log(1 + 1/A003095(n)^2)/2^n;
%o A076949   end if; return S;
%o A076949 end function;
%o A076949 SetDefaultRealField(RealField(120)); Exp(S(12)/2); // _G. C. Greubel_, Nov 29 2022
%o A076949 (SageMath)
%o A076949 @CachedFunction
%o A076949 def A003095(n): return 0 if (n==0) else 1 + A003095(n-1)^2
%o A076949 @CachedFunction
%o A076949 def S(n): return log(2)/2 if (n==1) else S(n-1) + log(1 + 1/(A003095(n))^2)/2^n
%o A076949 numerical_approx( exp(S(12)/2), digits=120) # _G. C. Greubel_, Nov 29 2022
%Y A076949 Cf. A003095, A077496, A213437, A258112, A258113.
%K A076949 cons,nonn
%O A076949 1,2
%A A076949 _Benoit Cloitre_, Nov 27 2002