A379408 a(n) = n + floor(n*s/r) + floor(n*t/r), where r = u^(1/4); s = u^(1/2); t = u^(3/4), u = golden ratio (A001622).

3, 6, 9, 13, 16, 19, 22, 27, 30, 33, 36, 40, 43, 46, 50, 54, 57, 60, 64, 67, 70, 73, 77, 81, 84, 88, 91, 94, 97, 101, 104, 108, 111, 115, 118, 121, 125, 128, 131, 135, 139, 142, 145, 148, 152, 155, 159, 163, 166, 169, 172, 176, 179, 182, 186, 190, 193, 196

Offset: 1

Author

Clark Kimberling, Jan 15 2025

Keywords

• nonn

Comments

This sequence and A379409 and A379410 partition the positive integers. For each k in A000027, write "a" if k=A379408(n) for some n, "b" if k=A379409(n) for some n, and "c" if k=A379410(n) for some n. Concatenating these letters for k = 1,2,3,... spells the following infinite word:

cbacbacbacbcabcabcabcacbcbacbacbacbacbcabcabcacbcabcbacbacbacbcabcabcacbacbcabcbacbacbcabcabcacbacbcabcabcb...

Crossrefs

Cf. A000027, A001622, A379409, A379410.

Programs

Mathematica

u = (1 + 5^(1/2))/2;
r = u^(1/4); s = u^(1/2); t = u^(3/4);
Table[n + Floor[n*s/r] + Floor[n*t/r], {n, 1, 120}]  (* A379408 *)
Table[n + Floor[n*r/s] + Floor[n*t/s], {n, 1, 120}]  (* A379409 *)
Table[n + Floor[n*r/t] + Floor[n*s/t], {n, 1, 120}]  (* A379410 *)

Formula

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