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A245976 Decimal expansion of the number whose continued fraction is given by A245920 (limit-reverse of an infinite Fibonacci word).

%I A245976 #16 Apr 28 2024 11:01:43
%S A245976 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 A245976 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 A245976 3,6,2,4,3,4,5,7,7,8,6,7,1,3,1,1,6,8
%N A245976 Decimal expansion of the number whose continued fraction is given by A245920 (limit-reverse of an infinite Fibonacci word).
%C A245976 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 A245976 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 A245976 [2,1,2,1,2,2,1,2,1,2,...] =  2.72994419476785022907837430700599816738...
%t A245976 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 A245976 x1 = N[FromContinuedFraction[s], 100]
%t A245976 r1 = RealDigits[x1, 10]  (* A245975 *)
%t A245976 ans = Join[{s[[p[0] = pos = seqPosition2[s, #] - 1]]}, #] &[{s[[1]]}];
%t A245976 cfs = Table[s = Drop[s, pos - 1]; ans = Join[{s[[p[n] = pos = seqPosition2[s, #] - 1]]}, #] &[ans], {n, z}];
%t A245976 rcf = Last[Map[Reverse, cfs]]  (* A245920 *)
%t A245976 x2 = N[FromContinuedFraction[rcf], z]
%t A245976 r2 = RealDigits[x2, 10] (* this sequence *)
%Y A245976 Cf. A245920, A245975.
%K A245976 nonn,cons
%O A245976 1,1
%A A245976 _Clark Kimberling_ and _Peter J. C. Moses_, Aug 08 2014