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

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, 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, 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

Offset: 1

Author

Benoit Cloitre, Nov 27 2002

Keywords

• cons
• nonn

Examples

			1.2259024435287485386279474959130085213212293209696612823177009072552339975...
		

Links

• G. C. Greubel, Table of n, a(n) for n = 1..10000
• Stephan Wagner, Volker Ziegler, Irrationality of growth constants associated with polynomial recursions, arXiv:2004.09353 [math.NT], 2020.

Crossrefs

Cf. A003095, A077496, A213437, A258112, A258113.

Programs

Magma

function A003095(n)
  if n eq 0 then return 0;
  else return 1 + A003095(n-1)^2;
  end if; return A003095;
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)/2); // G. C. Greubel, Nov 29 2022

Mathematica

A003095[n_]:= A003095[n]= If[n==0, 0, 1 + A003095[n-1]^2];
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]/2], 10, 120][[1]] (* G. C. Greubel, Nov 29 2022 *)

SageMath

@CachedFunction
def A003095(n): return 0 if (n==0) else 1 + A003095(n-1)^2
@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)/2), digits=120) # G. C. Greubel, Nov 29 2022

Formula

Equals sqrt(A077496). - Vaclav Kotesovec, Dec 17 2014