A258112 -id:A258112 - cp's OEIS frontend

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

Benoit Cloitre, Nov 27 2002

			1.2259024435287485386279474959130085213212293209696612823177009072552339975...

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.

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

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

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 *)

@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

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

A258113 Decimal expansion of a constant related to A007660.

Original entry on oeis.org

1, 1, 1, 3, 0, 5, 7, 9, 7, 5, 9, 0, 2, 9, 3, 1, 9, 3, 2, 8, 5, 3, 5, 9, 7, 7, 0, 7, 1, 6, 7, 5, 8, 4, 9, 1, 9, 0, 6, 6, 0, 0, 1, 8, 1, 5, 1, 0, 1, 8, 6, 5, 2, 7, 2, 0, 1, 4, 3, 7, 9, 7, 2, 4, 2, 0, 6, 9, 2, 7, 7, 1, 7, 2, 9, 7, 9, 8, 8, 2, 5, 9, 3, 8, 1, 6, 0, 9, 3, 6, 1, 4, 5, 4, 4, 5, 9, 4, 3, 5, 2, 2, 3, 4, 5

Offset: 1

Vaclav Kotesovec, May 20 2015

			1.1130579759029319328535977071675849190660018151018652720143797242069...

Cf. A007660, A076949, A250309, A258112.

A007660 = RecurrenceTable[{a[1]==0, a[2]==N[1,200], a[n]==a[n-1]*a[n-2]+1},a[n],{n,1,30}]; Do[Print[N[Exp[c2]/.Solve[Table[Log[A007660[[n]]]==c1*((1-Sqrt[5])/2)^n + c2*((1+Sqrt[5])/2)^n, {n,k,k+1}]], 120][[1]]],{k, Length[A007660]-2, Length[A007660]-1}];

Equals limit n->infinity (A007660(n))^((2/(1+sqrt(5)))^n).

A001056 a(n) = a(n-1)*a(n-2) + 1, a(0) = 1, a(1) = 3.

Original entry on oeis.org

1, 3, 4, 13, 53, 690, 36571, 25233991, 922832284862, 23286741570717144243, 21489756930695820973683319349467, 500426416062641238759467086706254193219790764168482, 10754042042885415070816603338436200915110904821126871858491675028294447933424899095

Offset: 0

N. J. A. Sloane, R. K. Guy

Archimedeans Problems Drive, Eureka, 19 (1957), 13.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

T. D. Noe, Table of n, a(n) for n = 0..17
A. V. Aho and N. J. A. Sloane, Some doubly exponential sequences, Fibonacci Quarterly, Vol. 11, No. 4 (1973), pp. 429-437.
A. V. Aho and N. J. A. Sloane, Some doubly exponential sequences, Fibonacci Quarterly, Vol. 11, No. 4 (1973), pp. 429-437 (original plus references that F.Q. forgot to include - see last page!)
Index entries for sequences of form a(n+1)=a(n)^2 + ...

Cf. A001622 (phi), A258112.

a:=[1,3];; for n in [3..13] do a[n]:=a[n-1]*a[n-2]+1; od; a; # G. C. Greubel, Sep 19 2019

a001056 n = a001056_list !! n
a001056_list = 1 : 3 : (map (+ 1 ) $
               zipWith (*) a001056_list $ tail a001056_list)
-- Reinhard Zumkeller, Aug 15 2012

I:=[1,3]; [n le 2 select I[n] else Self(n-1)*Self(n-2) + 1: n in [1..13]]; // G. C. Greubel, Sep 19 2019

a:= proc (n) option remember;
if n=0 then 1
elif n=1 then 3
else a(n-1)*a(n-2) + 1
end if
end proc;
seq(a(n), n = 0..13); # G. C. Greubel, Sep 19 2019

RecurrenceTable[{a[0]==1,a[1]==3,a[n]==a[n-1]*a[n-2]+1},a,{n,0,14}] (* Harvey P. Dale, Jul 17 2011 *)
t = {1, 3}; Do[AppendTo[t, t[[-1]] * t[[-2]] + 1], {n, 2, 14}] (* T. D. Noe, Jun 25 2012 *)

m=13; v=concat([1,3], vector(m-2)); for(n=3, m, v[n]=v[n-1]*v[n-2] +1 ); v \\ G. C. Greubel, Sep 19 2019

def a(n):
    if (n==0): return 1
    elif (n==1): return 3
    else: return a(n-1)*a(n-2) + 1
[a(n) for n in (0..13)] # G. C. Greubel, Sep 19 2019

a(n) ~ c^(phi^n), where c = A258112 = 1.7978784900091604813559508837..., phi = (1+sqrt(5))/2 = A001622. - Vaclav Kotesovec, Dec 17 2014

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

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A258113 Decimal expansion of a constant related to A007660.

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A001056 a(n) = a(n-1)*a(n-2) + 1, a(0) = 1, a(1) = 3.

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