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A105670 a(1)=1 then bracketing n by powers of 2 as f(t)=2^t for f(t) < n <= f(t+1), a(n) = f(t+1) - a(n-f(t)).

%I A105670 #30 Mar 16 2025 21:58:42
%S A105670 1,1,3,3,7,7,5,5,15,15,13,13,9,9,11,11,31,31,29,29,25,25,27,27,17,17,
%T A105670 19,19,23,23,21,21,63,63,61,61,57,57,59,59,49,49,51,51,55,55,53,53,33,
%U A105670 33,35,35,39,39,37,37,47,47,45,45,41,41,43,43,127,127,125,125,121,121,123
%N A105670 a(1)=1 then bracketing n by powers of 2 as f(t)=2^t for f(t) < n <= f(t+1), a(n) = f(t+1) - a(n-f(t)).
%H A105670 Vincenzo Librandi, <a href="/A105670/b105670.txt">Table of n, a(n) for n = 1..1000</a>
%F A105670 a(2n-1) = a(2n).
%F A105670 a(n) = 2*a(ceiling(n/2)) -1 + 2*t(ceiling(n/2)-1) where t(n) = A010060(n) is the Thue-Morse sequence.
%F A105670 a(2n-1) = a(2n) = 2*A006068(n-1)+1. - _Jeffrey Shallit_, Mar 15 2025
%p A105670 A062383 := proc(n)
%p A105670         ceil(log(n)/log(2)) ;
%p A105670         2^% ;
%p A105670 end proc:
%p A105670 A105670 := proc(n)
%p A105670         option remember;
%p A105670         if n = 1 then
%p A105670                 1;
%p A105670         else
%p A105670                 fn1 := A062383(n) ;
%p A105670                 fn := fn1/2 ;
%p A105670                 fn1-procname(n-fn) ;
%p A105670         end if;
%p A105670 end proc:
%p A105670 seq(A105670(n),n=1..80) ; # _R. J. Mathar_, Nov 06 2011
%t A105670 t[0] = 0; t[1] = 1; t[n_?EvenQ] := t[n] = t[n/2]; t[n_?OddQ] := t[n] = 1 - t[(n-1)/2]; a[1] = 1; a[n_?EvenQ] := a[n] = a[n - 1]; a[n_] := a[n] = 2*a[Ceiling[n/2]] - 1 + 2*t[Ceiling[n/2] - 1]; Table[a[n], {n, 1, 71}] (* _Jean-François Alcover_, Aug 13 2013 *)
%o A105670 (PARI) b(n,m)=if(n<2,1,m*m^floor(log(n-1)/log(m))-b(n-m^floor(log(n-1)/log(m)),m))
%Y A105670 Cf. A006068, A105669, A105672, A093347, A093348.
%K A105670 nonn,easy
%O A105670 1,3
%A A105670 _Benoit Cloitre_, May 03 2005
%E A105670 Typo in data corrected by _Jean-François Alcover_, Aug 13 2013