A189678 -id:A189678 - cp's OEIS frontend

A189679 n+[nr/t]+[ns/t]; r=1, s=arctan(1/2), t=arctan(2).

1, 3, 6, 8, 11, 13, 15, 18, 20, 23, 24, 27, 29, 31, 34, 36, 39, 41, 43, 46, 47, 50, 52, 55, 57, 59, 62, 64, 67, 69, 70, 73, 75, 78, 80, 83, 85, 87, 90, 92, 95, 96, 99, 101, 103, 106, 108, 111, 113, 115, 118, 119, 122, 124, 127, 129, 131, 134, 136, 139, 141, 142, 145, 147, 150, 152, 155, 157, 159, 162, 164, 167, 168, 170, 173, 175, 178, 180, 183, 185, 187, 190, 191, 194

Offset: 1

Clark Kimberling, Apr 25 2011

nonn

See A189677.

Cf. A189677, A189678.

(See A189677.)
Table[n+Floor[n/ArcTan[2]]+Floor[(n*ArcTan[1/2])/ArcTan[2]],{n,200}] (* Harvey P. Dale, Apr 07 2025 *)

A189677 n+[ns/r]+[nt/r]; r=1, s=arctan(1/2), t=arctan(2).

Original entry on oeis.org

2, 4, 7, 9, 12, 14, 17, 19, 22, 25, 28, 30, 33, 35, 37, 40, 42, 45, 48, 51, 53, 56, 58, 61, 63, 66, 68, 71, 74, 76, 79, 81, 84, 86, 89, 91, 94, 97, 100, 102, 105, 107, 109, 112, 114, 117, 120, 123, 125, 128, 130, 133, 135, 138, 140, 143, 146, 148, 151, 153, 156, 158, 161, 163, 166, 169, 172, 174, 176, 179, 181, 184, 186, 189, 192, 195, 197, 200, 202, 205, 207, 210, 212, 215

Offset: 1

Clark Kimberling, Apr 25 2011

nonn

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=1, s=arctan(1/2), t=arctan(2) gives
a=A189677, b=A189678, c=A189679.

Cf. A189678, A189679.

r=1; s=ArcTan[1/2]; t=ArcTan[2];
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}]  (*A189677*)
Table[b[n], {n, 1, 120}]  (*A189678*)
Table[c[n], {n, 1, 120}]  (*A189679*)

a(n)=n+[n*arctan(1/2)]+[n*arctan(2)].

cp's OEIS Frontend

A189679 n+[nr/t]+[ns/t]; r=1, s=arctan(1/2), t=arctan(2).

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