A190002 -id:A190002 - cp's OEIS frontend

A190004 A190002/2.

2, 4, 7, 9, 11, 14, 16, 19, 21, 23, 26, 28, 30, 33, 35, 38, 40, 42, 45, 47, 50, 52, 54, 57, 59, 61, 64, 66, 69, 71, 73, 76, 78, 80, 83, 85, 88, 90, 92, 95, 97, 100, 102, 104, 107, 109, 111, 114, 116, 119, 121, 123, 126, 128, 130, 133, 135, 138, 140, 142, 145, 147, 150, 152, 154, 157, 159, 161, 164, 166, 169, 171, 173, 176

Offset: 1

Clark Kimberling, May 03 2011

nonn

See A180002.

First differs from A182761 at n=55: a(55)=130, A182761(55)=131. - Bruno Berselli, Jun 04 2013

G. C. Greubel, Table of n, a(n) for n = 1..10000

Cf. A190002, A190003.

[(n + Floor(n*(Sinh(1))^2) + Floor(n*(Cosh(1))^2))/2: n in [1..100]]; // G. C. Greubel, Jan 11 2018

r=1; s=Sinh[1]^2; t=Cosh[1]^2;
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}]  (* A190002 *)
Table[b[n], {n, 1, 120}]  (* A190003 *)
Table[c[n], {n, 1, 120}]  (* A005408 *)
Table[a[n]/2, {n, 1, 120}](* A190004 *)
Table[b[n]/2, {n, 1, 120}](* A182760 *)

for(n=1,100, print1((n + floor(n*(sinh(1))^2) + floor(n*(cosh(1))^2))/2, ", ")) \\ G. C. Greubel, Jan 11 2018

A190002:  a(n) = n + [n*(sinh(1))^2] + [n*(cosh(1))^2].
A190003:  b(n) = n + [n*(csch(1))^2] + [n*(coth(1))^2].
A190004:  a(n)/2 = (n + [n*(sinh(1))^2] + [n*(cosh(1))^2])/2.
A005408:  c(n) = 2*n - 1.

A189999 a(n) = n + [ns/r] + [nt/r]; r=1, s=sinh(1), t=cosh(1).

Original entry on oeis.org

3, 7, 10, 14, 17, 22, 25, 29, 32, 36, 39, 44, 48, 51, 55, 58, 62, 66, 70, 73, 77, 80, 85, 89, 92, 96, 99, 103, 107, 111, 114, 118, 121, 125, 130, 133, 137, 140, 144, 148, 152, 155, 159, 162, 166, 170, 174, 178, 181, 185, 188, 193, 196, 200, 203, 207, 210, 215, 219, 222, 226, 229, 234, 237, 241, 244, 248, 251, 256, 260, 263, 267, 270

Offset: 1

Clark Kimberling, May 03 2011

nonn

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
a(n) = n + [n*s/r] + [n*t/r],
b(n) = n + [n*r/s] + [n*t/s],
c(n) = n + [n*r/t] + [n*s/t], where []=floor.
Taking r=1, s=sinh(1), t=cosh(1) gives
a=A189999, b=A190000, c=A190001.

G. C. Greubel, Table of n, a(n) for n = 1..10000

Cf. A190000, A190001, A190002.

[n + Floor(n*Sinh(1)) + Floor(n*Cosh(1)): n in [1..100]]; // G. C. Greubel, Jan 11 2018

r=1; s=Sinh[1]; t=Cosh[1];
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}]  (*A189999*)
Table[b[n], {n, 1, 120}]  (*A190000*)
Table[c[n], {n, 1, 120}]  (*A190001*)

for(n=1,100, print1(n + floor(n*sinh(1)) + floor(n*cosh(1)), ", ")) \\ G. C. Greubel, Jan 11 2018

A189999:  a(n) = n + [n*sinh(1)] + [n*cosh(1)].
A190000:  b(n) = n + [n*csch(1)] + [n*coth(1)].
A190001:  c(n) = n + [n*sech(1)] + [n*tanh(1)].

A190003 a(n) = n + [nr/s] + [nt/s]; r=1, s=(sinh(1))^2, t=(cosh(1))^2.

Original entry on oeis.org

2, 6, 10, 12, 16, 20, 24, 26, 30, 34, 36, 40, 44, 48, 50, 54, 58, 62, 64, 68, 72, 74, 78, 82, 86, 88, 92, 96, 98, 102, 106, 110, 112, 116, 120, 124, 126, 130, 134, 136, 140, 144, 148, 150, 154, 158, 162, 164, 168, 172, 174, 178, 182, 186, 188, 192, 196, 198, 202, 206, 210, 212, 216, 220, 224, 226, 230, 234, 236, 240

Offset: 1

Clark Kimberling, May 03 2011

nonn

See A190002.

G. C. Greubel, Table of n, a(n) for n = 1..10000

Cf. A190002, A182760.

[n + Floor(n/(Sinh(1))^2) + Floor(n/(Tanh(1))^2): n in [1..100]]; // G. C. Greubel, Jan 11 2018

seq(n+floor(n/s)+floor(n*t/s), n=1..100); # Robert Israel, Jan 12 2018

r=1; s=Sinh[1]^2; t=Cosh[1]^2;
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}]  (* A190002 *)
Table[b[n], {n, 1, 120}]  (* A190003 *)
Table[c[n], {n, 1, 120}]  (* A005408 *)
Table[a[n]/2, {n, 1, 120}](* A190004 *)
Table[b[n]/2, {n, 1, 120}](* A182760 *)

for(n=1,100, print1(n + floor(n/(sinh(1))^2) + floor(n/(tanh(1))^2), ", ")) \\ G. C. Greubel, Jan 11 2018

A190002:  a(n) = n + [n*(sinh(1))^2] + [n*(cosh(1))^2].
A190003:  b(n) = n + [n*(csch(1))^2] + [n*(coth(1))^2].
A005408:  c(n) = 2*n - 1.

cp's OEIS Frontend

A190004 A190002/2.

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A189999 a(n) = n + [ns/r] + [nt/r]; r=1, s=sinh(1), t=cosh(1).

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A190003 a(n) = n + [nr/s] + [nt/s]; r=1, s=(sinh(1))^2, t=(cosh(1))^2.

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cp's OEIS Frontend

A190004 A190002/2.

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Author

Keywords

Comments

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Crossrefs

Programs

Magma

Mathematica

PARI

Formula

A189999 a(n) = n + [n*s/r] + [n*t/r]; r=1, s=sinh(1), t=cosh(1).

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Author

Keywords

Comments

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Crossrefs

Programs

Magma

Mathematica

PARI

Formula

A190003 a(n) = n + [n*r/s] + [n*t/s]; r=1, s=(sinh(1))^2, t=(cosh(1))^2.

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Author

Keywords

Comments

Links

Crossrefs

Programs

Magma

Maple

Mathematica

PARI

Formula

A189999 a(n) = n + [ns/r] + [nt/r]; r=1, s=sinh(1), t=cosh(1).

A190003 a(n) = n + [nr/s] + [nt/s]; r=1, s=(sinh(1))^2, t=(cosh(1))^2.