A379420 a(n) = n + [n*s/r] + [n*t/r], where r = 5^(1/4); s = 5^(1/2); t = 5^(3/4) and [ ] = floor.

4, 8, 13, 17, 23, 27, 32, 36, 42, 46, 51, 55, 61, 65, 70, 74, 80, 84, 89, 93, 98, 103, 108, 112, 117, 122, 127, 131, 136, 141, 146, 150, 155, 160, 165, 169, 174, 178, 184, 188, 193, 197, 203, 207, 212, 216, 222, 226, 231, 235, 241, 245, 250, 254, 259, 264

Offset

1,1

Comments

This sequence and A379421 and A379422 partition the positive integers; see A184812 for a proof.

For each k in A000027, write "a" if k=A379420(n) for some n, "b" if k=A379421(n) for some n, and "c" if k=A379422(n) for some n. Concatenating these letters for k = 1,2,3,... spells the following infinite word:

cbcabccabcbcacbcabccbcabccabcbcacbcabccbcabccabcbcacbcabccbcabccabcbcacbcabccbcabccabcbcacbcabccbacbccabcbcacbcabccbacbccabcbca...

Links

Table of n, a(n) for n=1..56.

Formula

a(n) = n + [n*s/r] + [n*t/r], where r = 5^(1/4); s = 5^(1/2); t = 5^(3/4) and [ ] = floor.

a(n) = n + [n*r] + [n*r^2], where r = 5^(1/4) and [ ] = floor.

Mathematica

r = 5^(1/4); s = 5^(1/2); t = 5^(3/4);
Table[n + Floor[n*s/r] + Floor[n*t/r], {n, 1, 120}]  (* A379420 *)
Table[n + Floor[n*r/s] + Floor[n*t/s], {n, 1, 120}]  (* A379421 *)
Table[n + Floor[n*r/t] + Floor[n*s/t], {n, 1, 120}]  (* A379422 *)

Crossrefs

Cf. A184812, A379421, A379422.

Sequence in context: A189370 A311992 A311993 * A311994 A311995 A311996

Adjacent sequences: A379417 A379418 A379419 * A379421 A379422 A379423

Keyword

nonn

Author

Clark Kimberling, Jan 20 2025

Status

approved