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 A320831 Decimal expansion of the constant t having the continued fraction expansion {d(n), n>=0} such that the continued fraction expansion of 3*t yields partial denominators {4*d(n), n>=0}. 4
 1, 3, 7, 3, 7, 7, 4, 4, 1, 6, 4, 1, 1, 8, 9, 2, 7, 7, 9, 8, 3, 8, 5, 4, 9, 5, 0, 6, 6, 3, 9, 6, 0, 5, 3, 8, 3, 8, 0, 0, 7, 1, 3, 0, 0, 2, 6, 0, 7, 8, 4, 2, 1, 8, 9, 3, 2, 7, 9, 9, 0, 4, 0, 2, 4, 8, 2, 6, 7, 3, 3, 1, 5, 5, 2, 4, 1, 0, 4, 2, 5, 4, 5, 3, 6, 6, 8, 6, 9, 2, 3, 0, 9, 0, 7, 2, 9, 8, 0, 8, 9, 1, 3, 9, 8, 4, 8, 4, 5, 6, 5, 8, 1, 0, 0, 4, 2, 4, 4, 1, 1, 4, 1, 9, 3, 3, 0, 7, 2, 0, 4, 4, 7, 4, 5, 2, 1, 8, 6, 8, 4, 6, 4, 9, 6, 8, 7, 1, 3, 0 (list; constant; graph; refs; listen; history; text; internal format)
 OFFSET 1,2 COMMENTS Is this constant transcendental? 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, ...]. Further, let CF(x) denote the simple continued fraction expansion of x, then we have the related identities which hold for n >= 1: (C1) CF( (4*n+1) * sqrt((n+1)/n) ) = (4*n+3) * CF( sqrt((n+1)/n) ), (C2) CF( (2*n+1) * sqrt((n+2)/n) ) = (2*n+3) * CF( sqrt((n+2)/n) ). LINKS Paul D. Hanna, Table of n, a(n) for n = 1..3001 FORMULA Given t = [d(0); d(1), d(2), d(3), d(4), d(5), d(6), ..., A320834(n), ...], some related simple continued fractions are: (1) 3*t = [4*d(0); 4*d(1), 4*d(2), 4*d(3), 4*d(4), 4*d(5), ...], (2) 3*t/4 = [d(0); 16*d(1), d(2), 16*d(3), d(4), 16*d(5), d(6), ...], (3) 12*t = [16*d(0); d(1), 16*d(2), d(3), 16*d(4), d(5), 16*d(6), ...], (4) 3*t/2 = [2*d(0); 8*d(1), 2*d(2), 8*d(3), 2*d(4), 8*d(5), 2*d(6), ...], (5) 6*t = [8*d(0); 2*d(1), 8*d(2), 2*d(3), 8*d(4), 2*d(5), 8*d(6), ...]. EXAMPLE The decimal expansion of this constant t begins: t = 1.373774416411892779838549506639605383800713002607842189327990402... The simple continued fraction expansion of t begins: 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, ..., A320834(n), ...]. such that the simple continued fraction expansion of 3*t begins: 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*A320834(n), ...]. The initial 1000 digits in the decimal expansion of t are t = 1.37377441641189277983854950663960538380071300260784\ 21893279904024826733155241042545366869230907298089\ 13984845658100424411419330720447452186846496871309\ 93611537665995205176080835177406950895125756891119\ 80915134888807230285924622832070894390440105854297\ 09681249062135643627285425138701026540290422483564\ 88492730997578199229485083989821808079068883171984\ 07031141426574548209572058371283039520211641927528\ 87513532330143554990150517422840360714935764293437\ 57458872431528639369813728159265563110330718628799\ 96977704230617847144362021013013790221614910568231\ 27450894001787415768703790366780081874847582449346\ 46012195526919702308642260551245557225670489756926\ 10073356625047859682542328603265305024792731699661\ 07707346790590697402355533935869626197258295389474\ 34717509042975989479155252899700525467717059228960\ 29994752938551798255237161498633969001654212726415\ 44019114620209689305131806997427047626502096289167\ 37623579663973917818431418184110200421986541799594\ 29845246183090799528184841814574898135012164288045... ... The initial 1000 terms in the continued fraction expansion of t are 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, 24,2,1,2,2,2,1,2,24,1,2,1,24,1,2,2,2,1,2,24,2,1,2,2,2, 2,2,2,2,1,31,2,1,2,12,2,1,2,2,2,1,2,12,2,1,2,31,1,2,1, 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, 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, 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, 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, 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, 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, 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, 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, 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, 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, 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, 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, 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, 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, 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, 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, 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, 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, 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, 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, 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, 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, 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, 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, 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, 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, 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, 2,1,2,2,2,1,2,24,1,2,1,52,1,2,1,24,1,2,1,31,2,1,2,12,2, 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, 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, 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, 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, 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, 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, 1,2,2,2,1,2,12,2,1,2,31,1,2,2,2,2,2,2,2,1,2,24,1,2, ...]. ... GENERATING METHOD. Start with CF = [1] and repeat (PARI code): t = (1/3)*contfracpnqn(4*CF)[1,1]/contfracpnqn(4*CF)[2,1]; CF = contfrac(t) This method can be illustrated as follows. t0 = [1] = 1; t1 = (1/3)*[4] = [1; 3] = 4/3; t2 = (1/3)*[4; 12] = [1; 2, 1, 3, 3] = 49/36; t3 = (1/3)*[4; 8, 4, 12, 12] = [1; 2, 1, 2, 12, 4, 36] = 20116/14643; 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; 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; ... where this constant t equals the limit of the iterations of the above process. PROG (PARI) /* Generate over 3400 digits in the decimal expansion */ CF=[1]; {for(i=1, 12, t = (1/3)*contfracpnqn(4*CF)[1, 1]/contfracpnqn(4*CF)[2, 1]; CF = contfrac(t) ); } for(n=1, 150, print1(floor(10^(n-1)*t)%10, ", ")) CROSSREFS Cf. A320834, A320833, A320832, A320411. Sequence in context: A375152 A200481 A016665 * A120124 A151573 A113832 Adjacent sequences: A320828 A320829 A320830 * A320832 A320833 A320834 KEYWORD nonn,cons AUTHOR Paul D. Hanna, Oct 23 2018 STATUS approved

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Last modified August 8 10:02 EDT 2024. Contains 375019 sequences. (Running on oeis4.)