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

1,1

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...

Links

Table of n, a(n) for n=1..58.

Formula

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

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 *)

Crossrefs

Cf. A000027, A001622, A379409, A379410.

Sequence in context: A186516 A059540 A190363 * A059550 A080081 A171982

Adjacent sequences: A379405 A379406 A379407 * A379409 A379410 A379411

Keyword

nonn

Author

Clark Kimberling, Jan 15 2025

Status

approved