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A019587 The left budding sequence: # of i such that 0<i<=n and 0 < {tau*i} <= {tau*n}, where {} is fractional part. 7
1, 1, 3, 2, 1, 5, 3, 8, 5, 2, 9, 5, 1, 10, 5, 15, 9, 3, 15, 8, 21, 13, 5, 20, 11, 2, 19, 9, 27, 16, 5, 25, 13, 1, 23, 10, 33, 19, 5, 30, 15, 41, 25, 9, 37, 20, 3, 33, 15, 46, 27, 8, 41, 21, 55, 34, 13, 49, 27, 5, 43, 20, 59, 35, 11, 52, 27, 2, 45, 19, 63, 36, 9, 55, 27, 74, 45, 16, 65 (list; graph; refs; listen; history; text; internal format)
OFFSET
1,3
REFERENCES
J. H. Conway, personal communication.
LINKS
N. J. A. Sloane, Classic Sequences
FORMULA
a(n)+A194733(n)=n.
EXAMPLE
{r}=0.61...; {2r}=0.23...; {3r}=0.85...; {4r}=0.47...;
so that a(4)=2.
MAPLE
Digits := 100;
A019587 := proc(n::posint)
local a, k, phi, kfrac, nfrac ;
phi := (1+sqrt(5))/2 ;
a :=0 ;
nfrac := n*phi-floor(n*phi) ;
for k from 1 to n do
kfrac := k*phi-floor(k*phi) ;
if evalf(kfrac-nfrac) <= 0 then
a := a+1 ;
end if;
end do:
a ;
end proc:
seq(A019587(n), n=1..100) ; # R. J. Mathar, Aug 13 2021
MATHEMATICA
r = GoldenRatio; p[x_] := FractionalPart[x];
u[n_, k_] := If[p[k*r] <= p[n*r], 1, 0]
v[n_, k_] := If[p[k*r] > p[n*r], 1, 0]
s[n_] := Sum[u[n, k], {k, 1, n}]
t[n_] := Sum[v[n, k], {k, 1, n}]
Table[s[n], {n, 1, 100}] (* A019587 *)
Table[t[n], {n, 1, 100}] (* A194733 *)
(* Clark Kimberling, Sep 02 2011 *)
PROG
(Haskell)
a019587 n = length $ filter (<= nTau) $
map (snd . properFraction . (* tau) . fromInteger) [1..n]
where (_, nTau) = properFraction (tau * fromInteger n)
tau = (1 + sqrt 5) / 2
-- Reinhard Zumkeller, Jan 28 2012
CROSSREFS
Sequence in context: A113977 A183162 A309219 * A102427 A080883 A021315
KEYWORD
nonn,easy,nice
AUTHOR
EXTENSIONS
Extended by Ray Chandler, Apr 18 2009
STATUS
approved

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Last modified March 19 06:47 EDT 2024. Contains 370953 sequences. (Running on oeis4.)