A378142 a(n) = n + floor(n*s/r) + floor(n*t/r), where r=2^(1/4), s=2^(1/2), t=2^(3/4).

3, 6, 10, 13, 17, 21, 24, 28, 31, 35, 39, 42, 46, 49, 53, 57, 61, 64, 67, 71, 74, 79, 82, 85, 89, 92, 97, 100, 104, 107, 110, 115, 118, 122, 125, 128, 133, 136, 140, 143, 146, 150, 154, 158, 161, 165, 168, 172, 176, 179, 183, 186, 190, 194, 197, 201, 204

Offset: 1

Author

Clark Kimberling, Jan 13 2025

Keywords

• nonn

Comments

The sequences A378142, A378185, A379510, partition the positive integers (A000027), as proved at A184812:

A378142: 3,6,10,13,17,21,24,28,32,35,...

A378185: 2,5,8,11,14,18,20,23,26,29,,...

A379510: 1,4,7,9,12,15,16,19,22,25,27,...

For each integer k >= 1, write "a" if k=A378142(n) for some n, "b" if k=A378185(n) for some n, and "c" if k=A379510(n) for some n. Concatenating these letters for k = 1,2,3,... spells the following infinite word:

cbacbacbcabcabccabcbacbacbcabcacbcabcbacbacbcacbacbcabcbacbcabcacbacbcabcabcbcacbacbacbcabcabccbacbacb...

Crossrefs

Cf. A000027, A010767, A184812, A378185, A379510.

Programs

Mathematica

r=2^(1/4); s=2^(1/2); t=2^(3/4);
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}]  (* A378142 *)
Table[b[n], {n, 1, 120}]  (* A378185 *)
Table[c[n], {n, 1, 120}]  (* A379510 *)

Formula

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

Extensions

Name corrected by Clark Kimberling, Jan 20 2025