A378142 - OEIS

%I #14 Jan 20 2025 22:45:51

%S 3,6,10,13,17,21,24,28,31,35,39,42,46,49,53,57,61,64,67,71,74,79,82,

%T 85,89,92,97,100,104,107,110,115,118,122,125,128,133,136,140,143,146,

%U 150,154,158,161,165,168,172,176,179,183,186,190,194,197,201,204

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

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

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

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

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

%C 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:

%C cbacbacbcabcabccabcbacbacbcabcacbcabcbacbacbcacbacbcabcbacbcabcacbacbcabcabcbcacbacbacbcabcabccbacbacb...

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

%t r=2^(1/4); s=2^(1/2); t=2^(3/4);

%t a[n_]:=n+Floor[n*s/r]+Floor[n*t/r];

%t b[n_]:=n+Floor[n*r/s]+Floor[n*t/s];

%t c[n_]:=n+Floor[n*r/t]+Floor[n*s/t];

%t Table[a[n], {n, 1, 120}]  (* A378142 *)

%t Table[b[n], {n, 1, 120}]  (* A378185 *)

%t Table[c[n], {n, 1, 120}]  (* A379510 *)

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

%K nonn

%O 1,1

%A _Clark Kimberling_, Jan 13 2025

%E Name corrected by _Clark Kimberling_, Jan 20 2025