cp's OEIS Frontend

A360649 The exponents that occur in the greedy representation of 1/2 as a sum of powers of 2/3.

%I A360649 #24 Nov 04 2024 01:46:10
%S A360649 2,8,11,14,16,26,33,38,45,48,51,53,65,69,72,80,83,89,94,101,105,109,
%T A360649 115,118,123,131,139,142,148,152,157,160,164,170,176,179,182,185,188,
%U A360649 193,197,208,214,220,223,225,232,234,240,243,247,250,254,258,261,271
%N A360649 The exponents that occur in the greedy representation of 1/2 as a sum of powers of 2/3.
%C A360649 These numbers correspond to the 1's in the (3/2)-expansion of 1/2, as defined by Renyi.
%H A360649 W. Parry, <a href="https://doi.org/10.1007/BF02020954">On the beta-Expansions of Real Numbers</a>, Acta Math. Acad. Sci. Hungar. 11, 401-416, 1960.
%H A360649 A. Renyi, <a href="https://doi.org/10.1007/BF02020331">Representations for real numbers and their ergodic properties</a>, Acta Math. Acad. Sci. Hung. 8 (1957) 477-493.
%F A360649 a(n) = A077468(n+1) - 1. - _Andrey Zabolotskiy_, Nov 03 2024
%e A360649 The first power of 2/3 that is smaller than 1/2 is (2/3)^2, so the first term of the sequence is 2. Subtracting (2/3)^2 from 1/2 leaves 1/18. The first power of 2/3 that is less than 1/18 is (2/3)^8, so the next term of the sequence is 8.
%p A360649 x:= 1/2:
%p A360649 for i from 1 to 100 do
%p A360649   A[i]:= ceil(log[2/3](x));
%p A360649   x:= x-(2/3)^A[i];
%p A360649 od:
%p A360649 seq(A[i],i=1..100); # _Robert Israel_, Feb 15 2023
%t A360649 PositionIndex[RealDigits[1/2, 3/2, 100, -1][[1]]][[2]]
%Y A360649 Cf. A058840, A077468.
%K A360649 nonn
%O A360649 1,1
%A A360649 _James Propp_, Feb 15 2023