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The right budding sequence: # of i such that 0 < i <= n and {tau*n} <= {tau*i} < 1, where {} is fractional part.
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%I #33 Sep 03 2023 08:44:44

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

%D J. H. Conway, personal communication.

%H Reinhard Zumkeller, <a href="/A019588/b019588.txt">Table of n, a(n) for n = 1..1000</a>

%H N. J. A. Sloane, <a href="/classic.html#WYTH">Classic Sequences</a>

%F a(n) = A194733(n) + 1.

%t r = -GoldenRatio; p[x_] := FractionalPart[x];

%t u[n_, k_] := If[p[k*r] <= p[n*r], 1, 0]

%t v[n_, k_] := If[p[k*r] > p[n*r], 1, 0]

%t s[n_] := Sum[u[n, k], {k, 1, n}]

%t t[n_] := Sum[v[n, k], {k, 1, n}]

%t Table[s[n], {n, 1, 100}] (* A019588 *)

%t Table[t[n], {n, 1, 100}] (* A194734 *)

%t (* _Clark Kimberling_, Sep 02 2011 *)

%t 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 (Haskell)

%o a019588 n = length $ filter (nTau <=) $

%o map (snd . properFraction . (* tau) . fromInteger) [1..n]

%o where (_, nTau) = properFraction (tau * fromInteger n)

%o tau = (1 + sqrt 5) / 2

%o -- _Reinhard Zumkeller_, Jan 28 2012

%Y Cf. A019587, A194734, A194738.

%K nonn,easy,nice

%O 1,2

%A _N. J. A. Sloane_ and _J. H. Conway_

%E Extended by _Ray Chandler_, Apr 18 2009