A379411 a(n) = n + floor(n*s/r) + floor(n*t/r), where r = e^(1/4), s = e^(1/2), t = e^(3/4).

3, 7, 10, 15, 19, 22, 26, 31, 34, 38, 43, 46, 50, 54, 58, 62, 66, 70, 74, 77, 81, 86, 89, 93, 98, 101, 105, 109, 113, 117, 121, 125, 129, 133, 136, 141, 145, 148, 153, 156, 160, 164, 168, 172, 176, 180, 184, 188, 191, 196, 200, 203, 208, 212, 215, 219, 223

Offset: 1

Author

Clark Kimberling, Jan 18 2025

Keywords

• nonn

Comments

This sequence and A379412 and A379413 partition the positive integers; see A184812 for a proof. For each k in A000027, write "a" if k=A379411(n) for some n, "b" if k=A379412(n) for some n, and "c" if k=A379413(n) for some n. Concatenating these letters for k = 1,2,3,... spells the following infinite word:

cbacbcabcacbcbacbcabcacbcabcbcacbacbcabcbcacbacbcabccabcbacbcabccabcbacbcacbacbcabcbcacbacbcabccbacbacbcabccabcbacbcacbcabcbacbcacbcabcabccbacb...

Crossrefs

Cf. A184812, A379412, A379413.

Cf. A092042, A019774, A331501.

Programs

Mathematica

r = E^(1/4); s = E^(1/2); t = E^(3/4);
Table[n + Floor[n*s/r] + Floor[n*t/r], {n, 1, 120}]  (* A379411 *)
Table[n + Floor[n*r/s] + Floor[n*t/s], {n, 1, 120}]  (* A379412 *)
Table[n + Floor[n*r/t] + Floor[n*s/t], {n, 1, 120}]  (* A379413 *)

Formula

a(n) = n + floor(n*r) + floor(n*r^2), where r = e^(1/4).