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A019588 The right budding sequence: # of i such that 0 < i <= n and {tau*n} <= {tau*i} < 1, where {} is fractional part.

%I A019588 #40 Jul 11 2025 01:26:02
%S A019588 1,2,1,3,5,2,5,1,5,9,3,8,13,5,11,2,9,16,5,13,1,10,19,5,15,25,9,20,3,
%T A019588 15,27,8,21,34,13,27,5,20,35,11,27,2,19,36,9,27,45,16,35,5,25,45,13,
%U A019588 34,1,23,45,10,33,56,19,43,5,30,55,15,41,67,25,52,9,37,65,20,49,3,33,63,15
%N A019588 The right budding sequence: # of i such that 0 < i <= n and {tau*n} <= {tau*i} < 1, where {} is fractional part.
%C A019588 Also, the number of distinct blocks of the Fibonacci word (A003849) containing the maximum possible number of 1's for such a block. - _Jeffrey Shallit_, Jul 09 2025
%D A019588 J. H. Conway, personal communication.
%H A019588 Reinhard Zumkeller, <a href="/A019588/b019588.txt">Table of n, a(n) for n = 1..1000</a>
%H A019588 N. J. A. Sloane, <a href="/classic.html#WYTH">Classic Sequences</a>
%F A019588 a(n) = A194733(n) + 1.
%t A019588 r = -GoldenRatio; p[x_] := FractionalPart[x];
%t A019588 u[n_, k_] := If[p[k*r] <= p[n*r], 1, 0]
%t A019588 v[n_, k_] := If[p[k*r] > p[n*r], 1, 0]
%t A019588 s[n_] := Sum[u[n, k], {k, 1, n}]
%t A019588 t[n_] := Sum[v[n, k], {k, 1, n}]
%t A019588 Table[s[n], {n, 1, 100}]   (* A019588 *)
%t A019588 Table[t[n], {n, 1, 100}]   (* A194734 *)
%t A019588 (* _Clark Kimberling_, Sep 02 2011 *)
%t A019588 Fold[Join[#1, Range[#1[[#2]], Length[#1] + 1 + Floor[GoldenRatio (#2 + 1)] - Floor[GoldenRatio #2], #2 + 1]] &, {1, 2}, Range[30]] (* _Birkas Gyorgy_, May 24 2012 *)
%o A019588 (Haskell)
%o A019588 a019588 n = length $ filter (nTau <=) $
%o A019588             map (snd . properFraction . (* tau) . fromInteger) [1..n]
%o A019588    where (_, nTau) = properFraction (tau * fromInteger n)
%o A019588          tau = (1 + sqrt 5) / 2
%o A019588 -- _Reinhard Zumkeller_, Jan 28 2012
%Y A019588 Cf. A003849, A019587, A194734, A194738.
%K A019588 nonn,easy,nice
%O A019588 1,2
%A A019588 _N. J. A. Sloane_ and _J. H. Conway_
%E A019588 Extended by _Ray Chandler_, Apr 18 2009