%I #29 Oct 26 2018 21:40:48
%S 1,2,1,2,12,2,1,2,15,1,2,2,2,2,2,2,2,1,19,2,2,2,2,2,1,2,24,2,1,2,2,2,
%T 1,2,24,1,2,1,24,1,2,2,2,1,2,24,2,1,2,2,2,2,2,2,2,1,31,2,1,2,12,2,1,2,
%U 2,2,1,2,12,2,1,2,31,1,2,1,24,1,2,1,31,2,2,2,2,2,1,2,24,1,2,1,2,288,2,1,2,1,24,2,1,2,2,2,1,2,24,2,1,2,2,2,2,2,41,24,1,2,1,2,144,2,1,2,1,24,2,1,2,2,2,2,2,2,2,1,15,2,1,2,12,2,1,2,41,12,2,1,2,1,288
%N Continued fraction of a constant t with partial denominators {a(n), n>=0} such that the continued fraction of 3*t yields partial denominators {4*a(n), n>=0}.
%C Is this constant transcendental?
%C Compare to the continued fraction expansions of sqrt(3) and 3*sqrt(3), which are related by a factor of 5: sqrt(3) = [1; 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, ...] and 3*sqrt(3) = [5; 5, 10, 5, 10, 5, 10, 5, 10, 5, 10, ...].
%C Further, let CF(x) denote the simple continued fraction expansion of x, then we have the related identities which hold for n >= 1:
%C (C1) CF( (4*n+1) * sqrt((n+1)/n) ) = (4*n+3) * CF( sqrt((n+1)/n) ),
%C (C2) CF( (2*n+1) * sqrt((n+2)/n) ) = (2*n+3) * CF( sqrt((n+2)/n) ).
%H Paul D. Hanna, <a href="/A320834/b320834.txt">Table of n, a(n) for n = 0..4000</a>
%F Given t = [a(0); a(1), a(2), a(3), a(4), a(5), a(6), ...], some related simple continued fractions are:
%F (1) 3*t = [4*a(0); 4*a(1), 4*a(2), 4*a(3), 4*a(4), 4*a(5), ...],
%F (2) 3*t/4 = [a(0); 16*a(1), a(2), 16*a(3), a(4), 16*a(5), a(6), ...],
%F (3) 12*t = [16*a(0); a(1), 16*a(2), a(3), 16*a(4), a(5), 16*a(6), ...],
%F (4) 3*t/2 = [2*a(0); 8*a(1), 2*a(2), 8*a(3), 2*a(4), 8*a(5), 2*a(6), ...],
%F (5) 6*t = [8*a(0); 2*a(1), 8*a(2), 2*a(3), 8*a(4), 2*a(5), 8*a(6), ...].
%e The decimal expansion of this constant t begins:
%e t = 1.373774416411892779838549506639605383800713002607842189327990402...
%e The simple continued fraction expansion of t begins:
%e t = [1; 2, 1, 2, 12, 2, 1, 2, 15, 1, 2, 2, 2, 2, 2, 2, 2, 1, 19, 2, 2, 2, 2, 2, 1, 2, 24, 2, 1, 2, 2, 2, 1, 2, 24, ... , a(n), ...].
%e such that the simple continued fraction expansion of 3*t begins:
%e 3*t = [4; 8, 4, 8, 48, 8, 4, 8, 60, 4, 8, 8, 8, 8, 8, 8, 8, 4, 76, 8, 8, 8, 8, 8, 4, 8, 96, 8, 4, 8, 8, 8, 4, 8, 96, ... , 4*a(n), ...].
%e The initial 1000 terms in the continued fraction expansion of t are
%e t = [1;2,1,2,12,2,1,2,15,1,2,2,2,2,2,2,2,1,19,2,2,2,2,2,1,2,
%e 24,2,1,2,2,2,1,2,24,1,2,1,24,1,2,2,2,1,2,24,2,1,2,2,2,
%e 2,2,2,2,1,31,2,1,2,12,2,1,2,2,2,1,2,12,2,1,2,31,1,2,1,
%e 24,1,2,1,31,2,2,2,2,2,1,2,24,1,2,1,2,288,2,1,2,1,24,2,1,
%e 2,2,2,1,2,24,2,1,2,2,2,2,2,41,24,1,2,1,2,144,2,1,2,1,24,
%e 2,1,2,2,2,2,2,2,2,1,15,2,1,2,12,2,1,2,41,12,2,1,2,1,288,
%e 1,2,1,2,12,41,2,1,2,24,2,1,2,2,2,1,2,12,2,1,2,31,1,2,1,
%e 24,1,2,1,2,3456,2,1,2,1,24,1,2,1,31,2,1,2,12,2,1,2,2,2,1,
%e 2,12,2,1,2,31,1,2,2,2,2,2,2,2,1,2,24,2,1,2,2,2,1,54,288,
%e 1,2,1,2,12,2,1,2,191,1,2,2,2,2,2,2,2,2,2,31,1,2,2,2,2,
%e 2,2,2,1,2,24,2,1,2,2,2,1,2,24,1,2,1,19,2,1,2,12,2,1,2,
%e 15,1,2,2,2,2,2,2,2,1,54,144,2,1,2,1,24,1,2,1,383,2,2,2,2,
%e 2,2,2,2,2,1,15,2,1,54,24,1,2,1,2,288,2,1,2,1,24,2,1,2,2,
%e 2,2,2,2,2,1,15,2,1,2,12,2,1,2,41,12,2,1,2,1,288,1,2,1,2,
%e 12,2,1,2,4607,1,2,2,2,2,2,2,2,2,2,31,1,2,1,24,1,2,1,40,1,
%e 2,2,2,2,2,2,2,1,15,2,1,2,12,2,1,2,2,2,1,2,12,2,1,2,15,
%e 1,2,2,2,2,2,2,2,1,40,1,2,1,24,2,1,2,2,2,1,2,24,2,1,2,
%e 2,2,2,2,2,2,1,31,2,1,2,12,2,1,2,2,2,1,2,12,72,3456,1,2,1,
%e 2,12,2,1,2,15,1,2,2,2,2,2,2,2,1,254,12,2,1,2,2,2,1,2,24,
%e 2,1,2,2,2,1,2,24,2,1,2,41,12,2,1,2,2,2,1,2,24,2,1,2,2,
%e 2,1,2,12,2,1,2,31,1,2,2,2,2,2,2,2,1,2,24,1,2,1,2,288,1,
%e 2,1,2,12,25,2,1,2,12,2,1,2,15,1,2,2,2,2,2,2,2,1,19,2,2,
%e 2,2,2,1,2,24,2,1,2,2,2,1,2,24,1,2,1,71,2,1,191,2,1,2,12,
%e 2,1,2,1,288,1,2,1,2,12,510,1,2,2,2,1,2,24,2,1,2,2,2,1,2,
%e 24,2,1,2,2,2,2,2,19,1,2,2,2,2,2,71,1,2,31,1,2,1,24,1,2,
%e 1,2,3456,2,1,2,1,24,1,2,1,31,2,1,2,12,2,1,2,2,2,1,2,24,2,
%e 1,2,2,2,1,2,12,20,24,1,2,1,2,144,2,1,2,1,24,54,1,2,15,1,2,
%e 2,2,2,2,2,2,2,2,383,1,2,1,24,1,2,1,2,144,2,1,2,1,24,6142,1,
%e 2,1,24,2,1,2,2,2,1,2,24,2,1,2,2,2,1,2,24,41,2,2,2,2,2,
%e 2,2,31,1,2,1,24,1,2,1,52,1,2,1,24,2,1,2,2,2,1,2,24,2,1,
%e 2,2,2,2,2,19,1,2,2,2,2,2,2,2,1,15,2,1,2,12,2,1,2,2,2,
%e 1,2,12,2,1,2,15,1,2,2,2,2,2,2,2,1,19,2,2,2,2,2,1,2,24,
%e 2,1,2,2,2,1,2,24,1,2,1,52,1,2,1,24,1,2,1,31,2,1,2,12,2,
%e 1,2,2,2,1,2,12,2,1,2,31,1,2,2,2,2,2,2,2,1,2,24,2,1,2,
%e 2,2,1,2,24,1,2,1,40,1,2,2,2,2,2,2,2,1,15,2,1,2,12,2,1,
%e 2,2,2,1,2,12,2,1,2,15,1,2,95,1,2,4607,1,2,1,24,1,2,1,2,144,
%e 2,1,2,1,24,20,12,2,1,2,2,2,1,2,24,2,1,2,2,2,1,2,12,338,1,
%e 2,15,1,2,2,2,2,2,2,2,1,2,24,1,2,1,2,288,2,1,2,1,24,2,1,
%e 2,2,2,2,2,2,2,1,31,2,1,2,12,2,1,2,54,2,1,15,2,1,2,12,2,
%e 1,2,2,2,1,2,12,2,1,2,31,1,2,2,2,2,2,2,2,1,2,24,1,2, ...].
%e ...
%e The initial 1000 digits in the decimal expansion of t are
%e t = 1.37377441641189277983854950663960538380071300260784\
%e 21893279904024826733155241042545366869230907298089\
%e 13984845658100424411419330720447452186846496871309\
%e 93611537665995205176080835177406950895125756891119\
%e 80915134888807230285924622832070894390440105854297\
%e 09681249062135643627285425138701026540290422483564\
%e 88492730997578199229485083989821808079068883171984\
%e 07031141426574548209572058371283039520211641927528\
%e 87513532330143554990150517422840360714935764293437\
%e 57458872431528639369813728159265563110330718628799\
%e 96977704230617847144362021013013790221614910568231\
%e 27450894001787415768703790366780081874847582449346\
%e 46012195526919702308642260551245557225670489756926\
%e 10073356625047859682542328603265305024792731699661\
%e 07707346790590697402355533935869626197258295389474\
%e 34717509042975989479155252899700525467717059228960\
%e 29994752938551798255237161498633969001654212726415\
%e 44019114620209689305131806997427047626502096289167\
%e 37623579663973917818431418184110200421986541799594\
%e 29845246183090799528184841814574898135012164288045...
%e GENERATING METHOD.
%e Start with CF = [1] and repeat (PARI code):
%e t = (1/3)*contfracpnqn(4*CF)[1,1]/contfracpnqn(4*CF)[2,1]; CF = contfrac(t)
%e This method can be illustrated as follows.
%e t0 = [1] = 1 ;
%e t1 = (1/3)*[4] = [1; 3] = 4/3 ;
%e t2 = (1/3)*[4; 12] = [1; 2, 1, 3, 3] = 49/36 ;
%e t3 = (1/3)*[4; 8, 4, 12, 12] = [1; 2, 1, 2, 12, 4, 36] = 20116/14643 ;
%e t4 = (1/3)*[4; 8, 4, 8, 48, 16, 144] = [1; 2, 1, 2, 12, 2, 1, 2, 15, 1, 2, 5, 432] = 124502344/90627939;
%e t5 = (1/3)*[4; 8, 4, 8, 48, 8, 4, 8, 60, 4, 8, 20, 1728] = [1; 2, 1, 2, 12, 2, 1, 2, 15, 1, 2, 2, 2, 2, 2, 2, 2, 1, 19, 2, 2, 2, 2, 2, 1, 6, 5184] = 1018841077754176/741636374635107 ; ...
%e where this constant t equals the limit of the iterations of the above process.
%e The above method generates terms with exponential growth; the number of terms with each iteration begins:
%e [1, 2, 5, 7, 13, 27, 59, 119, 231, 463, 913, 1817, 3565, 7033, 13989, 27658, ...].
%o (PARI) /* Generate over 3500 terms in the continued fraction */
%o CF=[1];
%o {for(i=1, 12, t = (1/3)*contfracpnqn(4*CF)[1, 1]/contfracpnqn(4*CF)[2, 1];
%o CF = contfrac(t) ); CF}
%Y Cf. A320831, A320832, A320833, A320410, A320952.
%K nonn,cofr
%O 0,2
%A _Paul D. Hanna_, Oct 23 2018