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A114551 Continued fraction expansion of the constant (A114550) equal to Sum_{n>=0} 1/A112373(n) such that the partial quotients satisfy a(2n) = A112373(n) for n > 0 and a(2n+1) = A112373(n+1)/A112373(n) for n >= 0.

%I A114551 #23 Oct 03 2021 18:54:34
%S A114551 2,1,1,2,2,6,12,78,936,73086,68408496,4999703411742,
%T A114551 342022190843338960032,1710009514450915230711940280907486,
%U A114551 584861200495456320274313200204390612579749188443599552
%N A114551 Continued fraction expansion of the constant (A114550) equal to Sum_{n>=0} 1/A112373(n) such that the partial quotients satisfy a(2n) = A112373(n) for n > 0 and a(2n+1) = A112373(n+1)/A112373(n) for n >= 0.
%C A114551 A112373 is defined by the recurrence: let b(n) = A112373(n), then
%C A114551 b(n) = (b(n-1)^3 + b(n-1)^2)/b(n-2) for n >= 2 with b(0)=b(1)=1.
%H A114551 Michael De Vlieger, <a href="/A114551/b114551.txt">Table of n, a(n) for n = 0..20</a>
%H A114551 Taras Goy and Mark Shattuck, <a href="https://digitalcommons.pvamu.edu/aam/vol16/iss1/10/">Determinant Formulas of Some Hessenberg Matrices with Jacobsthal Entries Jacobsthal Entries</a>, Applications and Appl. Math. (2021) Vol. 16, Issue 1, Art. 10.
%H A114551 A. 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>.
%F A114551 a(2n) = a(2n-1)*a(2n-2) for n>=2, a(2n+1) = a(2n)*a(2n-1) + a(2n-1) for n>=1, with a(0)=2, a(1)=a(2)=1. - _Jeffrey Shallit_
%e A114551 2.584401724019776724812076147153331342112382090467969...
%e A114551 = Sum_{n>=0} 1/A112373(n) = 1/1 + 1/1 + 1/2 + 1/12 + 1/936 + 1/68408496 + ...
%e A114551 = [2;1,1,2,2,6,12,78,936,73086,68408496,...] (continued fraction).
%e A114551 The recurrence of partial quotients is demonstrated by:
%e A114551 (odd-index) a(7) = 78 = a(6)*a(5) + a(5) = 12*6 + 6;
%e A114551 (even-index) a(8) = 936 = a(7)*a(6) = 78*12.
%t A114551 a[0] = 2; a[1] = a[2] = 1;
%t A114551 a[n_] := a[n] = a[n-1] a[n-2] + Mod[n, 2] a[n-2];
%t A114551 a /@ Range[0, 14] (* _Jean-François Alcover_, Oct 01 2019 *)
%o A114551 (PARI) a(n)=if(n<0,0,if(n<3,[2,1,1][n+1],a(n-1)*a(n-2)+(n%2)*a(n-2)))
%Y A114551 Cf. A112373, A114550 (constant), A114552 (bisection).
%K A114551 cofr,nonn
%O A114551 0,1
%A A114551 _Paul D. Hanna_, Dec 08 2005