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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).

%I A378142 #14 Jan 20 2025 22:45:51
%S A378142 3,6,10,13,17,21,24,28,31,35,39,42,46,49,53,57,61,64,67,71,74,79,82,
%T A378142 85,89,92,97,100,104,107,110,115,118,122,125,128,133,136,140,143,146,
%U A378142 150,154,158,161,165,168,172,176,179,183,186,190,194,197,201,204
%N 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).
%C A378142 The sequences A378142, A378185, A379510, partition the positive integers (A000027), as proved at A184812:
%C A378142 A378142: 3,6,10,13,17,21,24,28,32,35,...
%C A378142 A378185: 2,5,8,11,14,18,20,23,26,29,,...
%C A378142 A379510: 1,4,7,9,12,15,16,19,22,25,27,...
%C A378142 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 A378142 cbacbacbcabcabccabcbacbacbcabcacbcabcbacbacbcacbacbcabcbacbcabcacbacbcabcabcbcacbacbacbcabcabccbacbacb...
%F A378142 a(n) = n + [w*n] + [w^2 n], where w = 2^(1/4) and [ ] = floor.
%t A378142 r=2^(1/4); s=2^(1/2); t=2^(3/4);
%t A378142 a[n_]:=n+Floor[n*s/r]+Floor[n*t/r];
%t A378142 b[n_]:=n+Floor[n*r/s]+Floor[n*t/s];
%t A378142 c[n_]:=n+Floor[n*r/t]+Floor[n*s/t];
%t A378142 Table[a[n], {n, 1, 120}]  (* A378142 *)
%t A378142 Table[b[n], {n, 1, 120}]  (* A378185 *)
%t A378142 Table[c[n], {n, 1, 120}]  (* A379510 *)
%Y A378142 Cf. A000027, A010767, A184812, A378185, A379510.
%K A378142 nonn
%O A378142 1,1
%A A378142 _Clark Kimberling_, Jan 13 2025
%E A378142 Name corrected by _Clark Kimberling_, Jan 20 2025