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A190002 a(n) = n + [n*s/r] + [n*t/r]; r=1, s=(sinh(1))^2, t=(cosh(1))^2.

%I A190002 #12 Sep 08 2022 08:45:56
%S A190002 4,8,14,18,22,28,32,38,42,46,52,56,60,66,70,76,80,84,90,94,100,104,
%T A190002 108,114,118,122,128,132,138,142,146,152,156,160,166,170,176,180,184,
%U A190002 190,194,200,204,208,214,218,222,228,232,238,242,246,252,256,260,266,270,276,280,284,290,294,300,304,308,314,318,322,328,332,338
%N A190002 a(n) = n + [n*s/r] + [n*t/r]; r=1, s=(sinh(1))^2, t=(cosh(1))^2.
%C A190002 This is one of three sequences that partition the positive integers.  In general, suppose that r, s, t are positive real numbers for which the sets {i/r: i>=1}, {j/s: j>=1}, {k/t: k>=1} are pairwise disjoint.  Let a(n) be the rank of n/r when all the numbers in the three sets are jointly ranked.  Define b(n) and c(n) as the ranks of n/s and n/t.  It is easy to prove that
%C A190002 a(n) = n + [n*s/r] + [n*t/r],
%C A190002 b(n) = n + [n*r/s] + [n*t/s],
%C A190002 c(n) = n + [n*r/t] + [n*s/t], where []=floor.
%C A190002 Taking r=1, s=(sinh(1))^2, t=(cosh(1))^2 gives
%C A190002 a=A190002, b=A190003, c=A190004.
%H A190002 G. C. Greubel, <a href="/A190002/b190002.txt">Table of n, a(n) for n = 1..10000</a>
%F A190002 A190002:  a(n) = n + [n*(sinh(1))^2] + [n*(cosh(1))^2].
%F A190002 A190003:  b(n) = n + [n*(csch(1))^2] + [n*(coth(1))^2].
%F A190002 A005408:  c(n) = 2*n - 1.
%t A190002 r=1; s=Sinh[1]^2; t=Cosh[1]^2;
%t A190002 a[n_] := n + Floor[n*s/r] + Floor[n*t/r];
%t A190002 b[n_] := n + Floor[n*r/s] + Floor[n*t/s];
%t A190002 c[n_] := n + Floor[n*r/t] + Floor[n*s/t];
%t A190002 Table[a[n], {n, 1, 120}]  (* A190002 *)
%t A190002 Table[b[n], {n, 1, 120}]  (* A190003 *)
%t A190002 Table[c[n], {n, 1, 120}]  (* A005408 *)
%t A190002 Table[a[n]/2, {n, 1, 120}](* A190004 *)
%t A190002 Table[b[n]/2, {n, 1, 120}](* A182760 *)
%o A190002 (PARI) for(n=1,100, print1(n + floor(n*(sinh(1))^2) + floor(n*(cosh(1))^2), ", ")) \\ _G. C. Greubel_, Jan 11 2018
%o A190002 (Magma) [n + Floor(n*(Sinh(1))^2) + Floor(n*(Cosh(1))^2): n in [1..100]]; // _G. C. Greubel_, Jan 11 2018
%Y A190002 Cf. A189999, A190003, A005408.
%K A190002 nonn
%O A190002 1,1
%A A190002 _Clark Kimberling_, May 03 2011