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A379510 a(n) = n + floor(n*r/t) + floor(n*s/t), where r=2^(1/4), s=2^(1/2), t=2^(3/4).

%I A379510 #11 Jan 13 2025 20:30:13
%S A379510 1,4,7,9,12,15,16,19,22,25,27,30,32,34,37,40,43,45,47,50,52,55,58,60,
%T A379510 63,65,68,70,73,76,78,80,83,86,88,91,94,95,98,101,103,106,109,111,113,
%U A379510 116,119,121,124,127,129,131,134,137,139,142,144,147,149,152
%N A379510 a(n) = n + floor(n*r/t) + floor(n*s/t), where r=2^(1/4), s=2^(1/2), t=2^(3/4).
%C A379510 The sequences A378142, A378185, A379510, partition the positive integers (A000027), as proved at A184812:
%C A379510 A378142: 3,6,10,13,17,21,24,28,32,35,...
%C A379510 A378185: 2,5,8,11,14,18,20,23,26,29,,...
%C A379510 A379510: 1,4,7,9,12,15,16,19,22,25,27,...
%C A379510 For each k in A000027, 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 A379510 cbacbacbcabcabccabcbacbacbcabcacbcabcbacbacbcacbacbcabcbacbcabcacbacbcabcabcbcacbacbacbcabcabccbacbacb...
%F A379510 a(n) = n + [n/w] + [n/w^2], where w = 2^(1/4) and [ ] = floor.
%t A379510 r=2^(1/4); s=2^(1/2); t=2^(3/4);
%t A379510 a[n_]:=n+Floor[n*s/r]+Floor[n*t/r];
%t A379510 b[n_]:=n+Floor[n*r/s]+Floor[n*t/s];
%t A379510 c[n_]:=n+Floor[n*r/t]+Floor[n*s/t];
%t A379510 Table[a[n], {n, 1, 120}]  (* A378142 *)
%t A379510 Table[b[n], {n, 1, 120}]  (* A378185 *)
%t A379510 Table[c[n], {n, 1, 120}]  (* A379510 *)
%Y A379510 Cf. A000027, A184812, A378142, A378185.
%K A379510 nonn
%O A379510 1,2
%A A379510 _Clark Kimberling_, Jan 13 2025