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%I #18 Oct 02 2019 05:19:00
%S 2,5,8,4,4,0,1,7,2,4,0,1,9,7,7,6,7,2,4,8,1,2,0,7,6,1,4,7,1,5,3,3,3,1,
%T 3,4,2,1,1,2,3,8,2,0,9,0,4,6,7,9,6,9,0,0,0,3,1,3,4,3,8,5,8,3,9,6,7,5,
%U 4,4,8,2,9,8,9,1,8,6,7,9,6,3,6,1,4,0,8,8,7,4,6,9,7,7,8,0,1,8,6,9,6,4,2,7,2
%N Decimal expansion of the constant Sum_{n>=0} 1/A112373(n), where the partial quotients of the continued fraction A114551 satisfy A114551(2n) = A112373(n) and A114551(2n+1) = A112373(n+1)/A112373(n).
%C A112373 is defined by the recurrence: let b(n) = A112373(n), then
%C b(n) =(b(n-1)^3 + b(n-1)^2)/b(n-2) for n>=2 with b(0)=b(1)=1.
%C Thus the sum of unit fractions 1/A112373(n) converges rapidly.
%H Andrew N. W. Hone, <a href="http://arxiv.org/abs/1507.00063">Curious continued fractions, nonlinear recurrences and transcendental numbers</a>, arXiv:1507.00063 [math.NT], 2015 and <a href="https://cs.uwaterloo.ca/journals/JIS/VOL18/Hone/hone3.html">J. Int. Seq. 18 (2015) # 15.8.4</a>.
%e 2.584401724019776724812076147153331342112382090467969...
%e = Sum_{n>=0} 1/A112373(n) = 1/1 +1/1 +1/2 +1/12 +1/936 +1/68408496 +...
%e = [2;1,1,2,2,6,12,78,936,73086,68408496,...] (continued fraction).
%t dm = 5; digits = 105;
%t b[n_] := b[n] = If[n < 2, 1, (b[n - 1]^3 + b[n - 1]^2)/b[n - 2]];
%t s[m_] := s[m] = N[Sum[1/b[n], {n, 0, m}], digits + 5];
%t s[m = dm];
%t s[m += dm];
%t While[RealDigits[s[m]] != RealDigits[s[m - dm]], m += dm];
%t RealDigits[s[m], 10, digits][[1]] (* _Jean-François Alcover_, Sep 30 2019 *)
%t c[0]=2; c[1] = c[2] = 1; c[n_] := c[n] = c[n-1] c[n-2] + Mod[n, 2] c[n-2];
%t RealDigits[FromContinuedFraction[c /@ Range[0, 14]], 10, 105][[1]] (* _Jean-François Alcover_, Oct 01 2019 *)
%Y Cf. A112373, A114551 (continued fraction), A114552.
%K cons,nonn
%O 1,1
%A _Paul D. Hanna_, Dec 08 2005
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