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 A320833 Decimal expansion of the constant t having the continued fraction expansion {d(n), n>=0} such that the continued fraction expansion of 2*t yields partial denominators {3*d(n), n>=0}. 5
 1, 6, 5, 1, 4, 9, 0, 4, 7, 4, 7, 3, 0, 8, 8, 1, 6, 3, 8, 0, 8, 6, 7, 9, 6, 8, 4, 4, 4, 4, 3, 0, 0, 6, 5, 7, 8, 6, 6, 6, 1, 3, 1, 9, 0, 6, 7, 6, 7, 9, 1, 2, 8, 7, 3, 7, 7, 1, 4, 8, 3, 0, 2, 7, 0, 8, 6, 1, 6, 9, 8, 0, 0, 7, 4, 4, 8, 6, 8, 1, 3, 1, 0, 6, 4, 3, 1, 0, 3, 9, 5, 1, 3, 7, 9, 4, 5, 1, 6, 7, 1, 5, 4, 1, 1, 4, 5, 0, 7, 6, 4, 7, 3, 6, 2, 3, 4, 6, 7, 3, 1, 6, 5, 5, 3, 4, 3, 7, 3, 6, 6, 7 (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), ..., A320832(n), ...], some related simple continued fractions are: (1) 2*t = [3*d(0); 3*d(1), 3*d(2), 3*d(3), 3*d(4), 3*d(5), ...], (2) 2*t/3 = [d(0); 9*d(1), d(2), 9*d(3), d(4), 9*d(5), d(6), ...], (3) 6*t = [9*d(0); d(1), 9*d(2), d(3), 9*d(4), d(5), 9*d(6), ...]. EXAMPLE The decimal expansion of this constant t begins: t = 1.6514904747308816380867968444430065786661319067679... The simple continued fraction expansion of t begins: t = [1; 1, 1, 1, 6, 1, 1, 1, 8, 1, 1, 1, 6, 1, 1, 1, 11, 1, 1, 1, 6, 1, 1, 1, 8, 1, 1, 1, 6, 1, 1, 1, 16, 6, ..., A320832(n), ...] such that the simple continued fraction expansion of 2*t begins: 2*t = [3; 3, 3, 3, 18, 3, 3, 3, 24, 3, 3, 3, 18, 3, 3, 3, 33, 3, 3, 3, 18, 3, 3, 3, 24, 3, 3, 3, 18, 3, 3, 3, 48, 18, ..., 3*A320832(n), ...]. ... The initial 1000 digits of constant t are t = 1.65149047473088163808679684444300657866613190676791\ 28737714830270861698007448681310643103951379451671\ 54114507647362346731655343736679236451104324525424\ 37390179560582497835710083063843943037795949756748\ 97440995664382615322263468614614477737059496505552\ 08331434371348884420941679386949556166027674330855\ 36885812669651061905701771944991773909674239753715\ 13170920756343463276796919694602222128457275152421\ 55344241016460665248526708333389356493782455903465\ 51804053948550486808342254050578114807877804220843\ 95652198152342146601232467945702955643471515688899\ 60645600818470508698165499450873578310068410378749\ 13752238554732608875401892185360771843689733221676\ 17468480327947937646720546426241804321256694683699\ 70360475351525492330828064189641281507963387604040\ 39551897256028156761617895973905334533561413827981\ 22417201150520589772826034062665207814244141469100\ 63982517271814133206125657700492407641669772124988\ 49178505556132383776229702780090744171002180938101\ 95839491920468860210551810985143159776520025263372... ... The initial 1000 terms of the simple continued fraction expansion of t are t = [1;1,1,1,6,1,1,1,8,1,1,1,6,1,1,1,11,1,1,1,6,1,1,1,8,1, 1,1,6,1,1,1,16,6,1,1,1,1,36,1,1,1,1,6,12,6,1,1,1,1,36, 1,1,1,1,6,24,36,1,1,1,1,6,1,1,1,53,1,1,1,6,1,1,1,1,36, 18,36,1,1,1,1,6,1,1,1,53,1,1,1,6,1,1,1,1,36,36,216,1,1,1, 1,6,1,1,1,8,1,1,1,6,1,1,1,79,6,1,1,1,1,36,1,1,1,1,6, 1,1,1,53,1,1,26,1,1,53,1,1,1,6,1,1,1,1,36,1,1,1,1,6,79, 1,1,1,6,1,1,1,8,1,1,1,6,1,1,1,1,216,54,1296,1,1,1,1,6,1, 1,1,8,1,1,1,6,1,1,1,11,1,1,1,6,1,1,1,8,1,1,1,6,1,1, 1,118,36,1,1,1,1,6,1,1,1,53,1,1,1,6,1,1,1,1,36,1,1,1,1, 6,79,1,1,1,6,39,6,1,1,1,79,6,1,1,1,1,36,1,1,1,1,6,1,1, 1,53,1,1,1,6,1,1,1,1,36,118,1,1,1,6,1,1,1,8,1,1,1,6,1, 1,1,11,1,1,1,6,1,1,1,8,1,1,1,6,1,1,1,1,1296,81,7776,1,1,1, 1,6,1,1,1,8,1,1,1,6,1,1,1,11,1,1,1,6,1,1,1,8,1,1,1, 6,1,1,1,16,6,1,1,1,1,36,1,1,1,1,6,12,6,1,1,1,1,36,1,1, 1,1,6,177,216,1,1,1,1,6,1,1,1,8,1,1,1,6,1,1,1,79,6,1,1, 1,1,36,1,1,1,1,6,1,1,1,53,1,1,1,6,1,1,1,1,36,118,1,1,1, 6,1,1,1,8,1,1,58,36,1,1,1,1,6,118,1,1,8,1,1,1,6,1,1,1, 1,216,1,1,1,1,6,1,1,1,8,1,1,1,6,1,1,1,79,6,1,1,1,1,36, 1,1,1,1,6,1,1,1,53,1,1,176,1,1,1,6,1,1,1,8,1,1,1,6,1, 1,1,11,1,1,1,6,1,1,1,8,1,1,1,6,1,1,1,16,6,1,1,1,1,36, 1,1,1,1,6,12,6,1,1,1,1,36,1,1,1,1,6,1,1,1,1943,1,1,121,46656, 1,1,1,1,6,1,1,1,8,1,1,1,6,1,1,1,11,1,1,1,6,1,1,1,8, 1,1,1,6,1,1,1,16,6,1,1,1,1,36,1,1,1,1,6,12,6,1,1,1,1, 36,1,1,1,1,6,24,36,1,1,1,1,6,1,1,1,53,1,1,1,6,1,1,1,1, 36,18,36,1,1,1,1,6,1,1,1,53,1,1,1,6,1,1,1,1,36,265,1,1,323, 1,1,1,6,1,1,1,1,36,1,1,1,1,6,12,6,1,1,1,1,36,1,1,1,1, 6,118,1,1,8,1,1,1,6,1,1,1,1,216,1,1,1,1,6,1,1,1,8,1,1, 1,6,1,1,1,79,6,1,1,1,1,36,1,1,1,1,6,1,1,1,53,1,1,176,1, 1,1,6,1,1,1,8,1,1,1,6,1,1,1,11,1,1,1,6,87,216,1,1,1,1, 6,1,1,1,8,1,1,176,1,1,1,6,12,6,1,1,1,1,36,1,1,1,1,6,1, 1,1,323,1,1,1,6,1,1,1,1,36,1,1,1,1,6,12,6,1,1,1,1,36,1, 1,1,1,6,118,1,1,8,1,1,1,6,1,1,1,1,216,1,1,1,1,6,1,1,1, 8,1,1,1,6,1,1,1,79,6,1,1,1,263,1,1,1,6,1,1,1,8,1,1,1, 6,1,1,1,11,1,1,1,6,1,1,1,8,1,1,1,6,1,1,1,16,6,1,1,1, 1,36,1,1,1,1,6,12,6,1,1,1,1,36,1,1,1,1,6,24,36,1,1,1,1, 6,1,1,1,53,1,1,1,6,1,1,1,1,36,18,36,1,1,1,1,6,1,1,1,53, 1,1,1,6,1,1,1,1,36,1,1,1,1,6,2914,1,1,1,6,181,1,1,69983,1,1, 1,6,1,1,1,1,36,1,1,1,1,6,12,6,1,1,1,1,36,1,1,1,1,6,16, 1,1,1,6,1,1,1,8,1,1,1,6,1,1,1,11,1,1,1,6,1,1,1,8,1, 1,1,6,1,1,1,23,1,1,8,1,1,1,6,1,1,1,1,216,1,1,1,1,6, ...]. ... GENERATING METHOD. Start with CF = [1] and repeat (PARI code): t = (1/2)*contfracpnqn(3*CF)[1,1]/contfracpnqn(3*CF)[2,1]; CF = contfrac(t) This method can be illustrated as follows. t0 = [1] = 1 ; t1 = (1/2)*[3] = [1; 2] = 3/2 ; t2 = (1/2)*[3; 6] = [1; 1, 1, 2, 2] = 19/12 ; t3 = (1/2)*[3; 3, 3, 6, 6] = [1, 1, 1, 1, 6, 3, 12] = 1281/776 ; t4 = (1/2)*[3; 3, 3, 3, 18, 9, 36] = [1; 1, 1, 1, 6, 1, 1, 1, 8, 1, 1, 4, 72] = 652299/394976 ; t5 = (1/2)*[3; 3, 3, 3, 18, 3, 3, 3, 24, 3, 3, 12, 216] = [1, 1, 1, 1, 6, 1, 1, 1, 8, 1, 1, 1, 6, 1, 1, 1, 11, 1, 1, 1, 6, 6, 432] = 43763571081/26499438992 ; ... where the limit of the above process approaches this constant t. PROG (PARI) /* Generate over 3000 digits in the decimal expansion */ CF=[1]; {for(i=1, 14, t = (1/2)*contfracpnqn(3*CF)[1, 1]/contfracpnqn(3*CF)[2, 1]; CF = contfrac(t) ); } for(n=1, 150, print1(floor(10^(n-1)*t)%10, ", ")) CROSSREFS Cf. A320832, A320831. Sequence in context: A372777 A296470 A195774 * A216582 A340543 A177824 Adjacent sequences: A320830 A320831 A320832 * A320834 A320835 A320836 KEYWORD nonn,cons AUTHOR Paul D. Hanna, Oct 22 2018 STATUS approved

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Last modified August 15 11:56 EDT 2024. Contains 375173 sequences. (Running on oeis4.)