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A189785 a(n) = n+floor(n*r/s)+floor(nt/s); r=Pi/2, s=arcsin(5/13), t=arcsin(12/13). 3
6, 14, 22, 30, 38, 46, 54, 62, 70, 78, 86, 94, 102, 110, 118, 126, 134, 142, 150, 158, 166, 174, 182, 190, 198, 206, 214, 222, 230, 238, 246, 254, 262, 270, 278, 286, 294, 302, 310, 318, 326, 334, 342, 350, 358, 366, 374, 380, 388, 396, 404, 412, 420, 428, 436, 444, 452, 460, 468, 476, 484, 492, 500, 508, 516, 524, 532, 540, 548 (list; graph; refs; listen; history; text; internal format)
OFFSET
1,1
COMMENTS
This is one of three sequences that partition the positive integers. In general, suppose that r, s, t are positive real numbers for which the sets {i/r: i>=1}, {j/s: j>=1}, {k/t: k>=1} are pairwise disjoint. Let a(n) be the rank of n/r when all the numbers in the three sets are jointly ranked. Define b(n) and c(n) as the ranks of n/s and n/t. It is easy to prove that
a(n)=n+[ns/r]+[nt/r],
b(n)=n+[nr/s]+[nt/s],
c(n)=n+[nr/t]+[ns/t], where []=floor.
Taking r=Pi/2, s=arcsin(5/13), t=arcsin(12/13) gives
a=A005408, b=A189785, c=A189786. Note that r=s+t.
a(n) first differs from A017137(n-1) at n=48 (a(48)=380 but A017137(47)=382). - Nathaniel Johnston, May 16 2011
LINKS
MATHEMATICA
r=Pi/2; s=ArcSin[5/13]; t=ArcSin[12/13];
a[n_] := n + Floor[n*s/r] + Floor[n*t/r];
b[n_] := n + Floor[n*r/s] + Floor[n*t/s];
c[n_] := n + Floor[n*r/t] + Floor[n*s/t];
Table[a[n], {n, 1, 120}] (*A005408*)
Table[b[n], {n, 1, 120}] (*A189785*)
Table[c[n], {n, 1, 120}] (*A189786*)
Table[b[n]/2, {n, 1, 120}] (*A189787*)
Table[c[n]/2, {n, 1, 120}] (*A004773*)
CROSSREFS
Sequence in context: A062316 A079299 A043445 * A017137 A211998 A190522
KEYWORD
nonn
AUTHOR
Clark Kimberling, Apr 27 2011
STATUS
approved

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Last modified February 20 18:38 EST 2024. Contains 370217 sequences. (Running on oeis4.)