A245976 - OEIS

%I #16 Apr 28 2024 11:01:43

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

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

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

%N Decimal expansion of the number whose continued fraction is given by A245920 (limit-reverse of an infinite Fibonacci word).

%C The (2,1)-version of the infinite Fibonacci word, A014675, as a sequence, is (2, 1, 2, 2, 1, 2, 1, 2, 2, 1, 2, 2, 1, 2, 1, 2, 2,...).  Its limit-reverse, A245920, is the sequence (2, 1, 2, 1, 2, 2, 1, 2, 1, 2, 2, 1, 2, 2, 1,...), which is the continued fraction for 2.729944...

%C For the (0,1)-version of the infinite Fibonacci word 0100101001001... (A003849), the decimal expansion is the same except for the first digit. That is 0.729944194... . - _Gandhar Joshi_, Mar 28 2024

%e [2,1,2,1,2,2,1,2,1,2,...] =  2.72994419476785022907837430700599816738...

%t z = 300; seqPosition2[list_, seqtofind_] := Last[Last[Position[Partition[list, Length[#], 1], Flatten[{___, #, ___}], 1, 2]]] &[seqtofind]; x = GoldenRatio;  s =  Differences[Table[Floor[n*x], {n, 1, z^2}]];  (* A014675 *)

%t x1 = N[FromContinuedFraction[s], 100]

%t r1 = RealDigits[x1, 10]  (* A245975 *)

%t ans = Join[{s[[p[0] = pos = seqPosition2[s, #] - 1]]}, #] &[{s[[1]]}];

%t cfs = Table[s = Drop[s, pos - 1]; ans = Join[{s[[p[n] = pos = seqPosition2[s, #] - 1]]}, #] &[ans], {n, z}];

%t rcf = Last[Map[Reverse, cfs]]  (* A245920 *)

%t x2 = N[FromContinuedFraction[rcf], z]

%t r2 = RealDigits[x2, 10] (* this sequence *)

%Y Cf. A245920, A245975.

%K nonn,cons

%O 1,1

%A _Clark Kimberling_ and _Peter J. C. Moses_, Aug 08 2014