A321092 - OEIS

%I #20 Nov 01 2018 06:47:08

%S 4,6,9,6,7,4,3,2,8,5,9,7,0,0,2,7,9,0,9,0,3,7,9,7,1,3,5,0,6,1,4,8,9,9,

%T 6,9,5,9,6,7,6,8,9,0,3,4,9,8,2,6,6,3,9,7,4,4,9,7,6,0,6,5,8,7,6,8,9,4,

%U 9,3,0,1,8,0,5,9,9,4,9,8,2,2,7,4,6,7,9,3,8,7,8,8,4,5,2,6,6,8,7,6,6,7,3,5,5,4,3,3,7,3,7,9,1,7,9,7,2,8,0,8,7,7,1,0,8,6,8,2,2,0,9,5,7,9,0,8,6,6,2,6,1,3,6,1,4,0,3,2,7,9,8,3,3,3,8,8,4,0,5,7,2,7,5,2,0,0,8,9,1,3,3,8,2,5,0,6,5,1,0,4,5,5,3,0,6,3,8,3,3,4,7,1,8,5,7,2,3,6,7,8,1

%N Decimal expansion of the constant z that satisfies: CF(3*z, n) = CF(z, n) + 10, for n >= 0, where CF(z, n) denotes the n-th partial denominator in the continued fraction expansion of z.

%H Paul D. Hanna, <a href="/A321092/b321092.txt">Table of n, a(n) for n = 1..7000</a>

%e The decimal expansion of this constant z begins:

%e z = 4.69674328597002790903797135061489969596768903498266...

%e The simple continued fraction expansion of z begins:

%e z = [4; 1, 2, 3, 2, 1, 3, 2, 1, 3, 1, 2, 3, 1, 2, 3, ..., A321091(n), ...];

%e such that the simple continued fraction expansion of 3*z begins:

%e 3*z = [14; 11, 12, 13, 12, 11, 13, 12, 11, 13, 11, 12, ..., A321091(n) + 10, ...].

%e EXTENDED TERMS.

%e The initial 1000 digits in the decimal expansion of z are

%e z = 4.69674328597002790903797135061489969596768903498266\

%e 39744976065876894930180599498227467938788452668766\

%e 73554337379179728087710868220957908662613614032798\

%e 33388405727520089133825065104553063833471857236781\

%e 30005222692602670820049366351838443544785254486817\

%e 36713432024147916330132466297271088195927148747751\

%e 61081134504127224293197514421215072180791056995531\

%e 25282254039576642227973626045065085745006810232418\

%e 31151864601858865871496634133860069172907530299184\

%e 80599130262819282015136761682297696200354791299160\

%e 72099154048193374507277756176907149849150248588944\

%e 75702959853076891940422757945105365738782828309624\

%e 89182729519410181321396021772327752921026193693551\

%e 52235778358918181495624102484144418903334090227672\

%e 68450214362152729231740655406007216125545132538964\

%e 63321922981643984915752295515263408732183582572996\

%e 74985150593020685391286450747231245540741605404683\

%e 64053241113305130300450809189365250050022411738651\

%e 75283281124343007394094179023437577924273245108697\

%e 96469643376214173186511094645179990191104328112759...

%e ...

%e The initial 1020 terms of the continued fraction of z are

%e Z = [4;1,2,3,2,1,3,2,1,3,1,2,3,1,2,3,2,1,3,1,2,3,1,2,3,2,1,3,1,2,3,

%e 2,1,3,2,1,3,1,2,3,2,1,3,2,1,3,1,2,3,1,2,3,2,1,3,1,2,3,2,1,3,

%e 2,1,3,1,2,3,2,1,3,2,1,3,1,2,3,1,2,3,2,1,3,1,2,3,2,1,3,2,1,3,

%e 1,2,3,1,2,3,2,1,3,1,2,3,1,2,3,2,1,3,1,2,3,2,1,3,2,1,3,1,2,3,

%e 1,2,3,2,1,3,1,2,3,1,2,3,2,1,3,1,2,3,2,1,3,2,1,3,1,2,3,2,1,3,

%e 2,1,3,1,2,3,1,2,3,2,1,3,1,2,3,2,1,3,2,1,3,1,2,3,1,2,3,2,1,3,

%e 1,2,3,1,2,3,2,1,3,1,2,3,2,1,3,2,1,3,1,2,3,1,2,3,2,1,3,1,2,3,

%e 1,2,3,2,1,3,1,2,3,2,1,3,2,1,3,1,2,3,2,1,3,2,1,3,1,2,3,1,2,3,

%e 2,1,3,1,2,3,2,1,3,2,1,3,1,2,3,1,2,3,2,1,3,1,2,3,1,2,3,2,1,3,

%e 1,2,3,2,1,3,2,1,3,1,2,3,2,1,3,2,1,3,1,2,3,1,2,3,2,1,3,1,2,3,

%e 2,1,3,2,1,3,1,2,3,2,1,3,2,1,3,1,2,3,1,2,3,2,1,3,1,2,3,2,1,3,

%e 2,1,3,1,2,3,1,2,3,2,1,3,1,2,3,1,2,3,2,1,3,1,2,3,2,1,3,2,1,3,

%e 1,2,3,2,1,3,2,1,3,1,2,3,1,2,3,2,1,3,1,2,3,2,1,3,2,1,3,1,2,3,

%e 2,1,3,2,1,3,1,2,3,1,2,3,2,1,3,1,2,3,2,1,3,2,1,3,1,2,3,1,2,3,

%e 2,1,3,1,2,3,1,2,3,2,1,3,1,2,3,2,1,3,2,1,3,1,2,3,1,2,3,2,1,3,

%e 1,2,3,1,2,3,2,1,3,1,2,3,2,1,3,2,1,3,1,2,3,2,1,3,2,1,3,1,2,3,

%e 1,2,3,2,1,3,1,2,3,2,1,3,2,1,3,1,2,3,1,2,3,2,1,3,1,2,3,1,2,3,

%e 2,1,3,1,2,3,2,1,3,2,1,3,1,2,3,2,1,3,2,1,3,1,2,3,1,2,3,2,1,3,

%e 1,2,3,2,1,3,2,1,3,1,2,3,2,1,3,2,1,3,1,2,3,1,2,3,2,1,3,1,2,3,

%e 2,1,3,2,1,3,1,2,3,1,2,3,2,1,3,1,2,3,1,2,3,2,1,3,1,2,3,2,1,3,

%e 2,1,3,1,2,3,2,1,3,2,1,3,1,2,3,1,2,3,2,1,3,1,2,3,2,1,3,2,1,3,

%e 1,2,3,2,1,3,2,1,3,1,2,3,1,2,3,2,1,3,1,2,3,2,1,3,2,1,3,1,2,3,

%e 1,2,3,2,1,3,1,2,3,1,2,3,2,1,3,1,2,3,2,1,3,2,1,3,1,2,3,1,2,3,

%e 2,1,3,1,2,3,1,2,3,2,1,3,1,2,3,2,1,3,2,1,3,1,2,3,2,1,3,2,1,3,

%e 1,2,3,1,2,3,2,1,3,1,2,3,2,1,3,2,1,3,1,2,3,1,2,3,2,1,3,1,2,3,

%e 1,2,3,2,1,3,1,2,3,2,1,3,2,1,3,1,2,3,2,1,3,2,1,3,1,2,3,1,2,3,

%e 2,1,3,1,2,3,2,1,3,2,1,3,1,2,3,2,1,3,2,1,3,1,2,3,1,2,3,2,1,3,

%e 1,2,3,2,1,3,2,1,3,1,2,3,1,2,3,2,1,3,1,2,3,1,2,3,2,1,3,1,2,3,

%e 2,1,3,2,1,3,1,2,3,1,2,3,2,1,3,1,2,3,1,2,3,2,1,3,1,2,3,2,1,3,

%e 2,1,3,1,2,3,2,1,3,2,1,3,1,2,3,1,2,3,2,1,3,1,2,3,2,1,3,2,1,3,

%e 1,2,3,1,2,3,2,1,3,1,2,3,1,2,3,2,1,3,1,2,3,2,1,3,2,1,3,1,2,3,

%e 1,2,3,2,1,3,1,2,3,1,2,3,2,1,3,1,2,3,2,1,3,2,1,3,1,2,3,2,1,3,

%e 2,1,3,1,2,3,1,2,3,2,1,3,1,2,3,2,1,3,2,1,3,1,2,3,1,2,3,2,1,3,

%e 1,2,3,1,2,3,2,1,3,1,2,3,2,1,3,2,1,3,1,2,3,2,1,3,2,1,3,1,2,3, ...].

%e Is there a pattern to the above terms?

%e ...

%e GENERATING METHOD.

%e Start with CF = [4] and repeat (PARI code):

%e {M = contfracpnqn(CF + vector(#CF,i, 10));

%e z = (1/3)*M[1,1]/M[2,1]; CF = contfrac(z)}

%e This method can be illustrated as follows.

%e z0 = [4] = 4;

%e z1 = (1/3)*[14] = [4; 1, 2] = 14/3;

%e z2 = (1/3)*[14; 11, 12] = [4, 1, 2, 3, 2, 1, 3, 3] = 1874/399;

%e z3 = (1/3)*[14; 11, 12, 13, 12, 11, 13, 13] = [4; 1, 2, 3, 2, 1, 3, 2, 1, 3, 1, 2, 3, 1, 2, 3, 2, 1, 3, 1, 2, 4] = 187227588/39863279;

%e z4 = (1/3)*[14; 11, 12, 13, 12, 11, 13, 12, 11, 13, 11, 12, 13, 11, 12, 13, 12, 11, 13, 11, 12, 14] = [4; 1, 2, 3, 2, 1, 3, 2, 1, 3, 1, 2, 3, 1, 2, 3, 2, 1, 3, 1, 2, 3, 1, 2, 3, 2, 1, 3, 1, 2, 3, 2, 1, 3, 2, 1, 3, 1, 2, 3, 2, 1, 3, 2, 1, 3, 1, 2, 3, 1, 2, 3, 2, 1, 3, 1, 2, 3, 2, 1, 3, 2, 1, 4] = 254413274852180460063336/54168017999228580539039; ...

%e where this constant z equals the limit of the iterations of the above process.

%o (PARI) /* Generate over 3600 digits */

%o {CF=[4]; for(i=1,8, M = contfracpnqn( CF + vector(#CF,i,10) ); z = (1/3)*M[1,1]/M[2,1]; CF = contfrac(z) )}

%o for(n=1,200,print1(floor(10^(n-1)*z)%10,", "))

%Y Cf. A321090, A321091, A321093, A321094, A321095, A321096, A321097, A321098.

%K nonn,cons

%O 1,1

%A _Paul D. Hanna_, Oct 27 2018