A182760 -id:A182760 - 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.

A182765 Beatty sequence for (6 + sqrt(2))/4.

Original entry on oeis.org

1, 3, 5, 7, 9, 11, 12, 14, 16, 18, 20, 22, 24, 25, 27, 29, 31, 33, 35, 37, 38, 40, 42, 44, 46, 48, 50, 51, 53, 55, 57, 59, 61, 63, 64, 66, 68, 70, 72, 74, 75, 77, 79, 81, 83, 85, 87, 88, 90, 92, 94, 96, 98, 100, 101, 103, 105, 107, 109, 111, 113, 114, 116, 118, 120, 122, 124, 126, 127

Offset: 1

Clark Kimberling, Nov 29 2010

nonn

Let u=(1+sqrt(2))/2 and v=sqrt(2). Jointly rank {ju} and {kv} as in the first comment at A182760; a(n) is the position of nu.

Cf. A182766.

Table[Floor[(6 + Sqrt@ 2) n/4], {n, 70}] (* Michael De Vlieger, Jun 23 2016 *)

A182765(n)=n*(6+sqrt(2))\4 \\ Requires sufficient realprecision (but the 64-bit default is enough up to n = 10^38). - M. F. Hasler, Jun 23 2016

a(n) = floor(r*n), where r = (6 + sqrt(2))/4.

a(n) = 2*n - 1 - floor(n/7) for n < 41, but this fails for a(41) = 75 onwards. - M. F. Hasler, Jun 23 2016

A182766 Beatty sequence for 5 - 2*sqrt(2).

Original entry on oeis.org

2, 4, 6, 8, 10, 13, 15, 17, 19, 21, 23, 26, 28, 30, 32, 34, 36, 39, 41, 43, 45, 47, 49, 52, 54, 56, 58, 60, 62, 65, 67, 69, 71, 73, 76, 78, 80, 82, 84, 86, 89, 91, 93, 95, 97, 99, 102, 104, 106, 108, 110, 112, 115, 117, 119, 121, 123, 125, 128, 130, 132, 134, 136, 138, 141, 143, 145, 147, 149, 152, 154, 156, 158, 160, 162, 165, 167

Offset: 1

Clark Kimberling, Nov 29 2010

nonn

Let u=(1+sqrt(2))/2 and v=sqrt(2). Jointly rank {ju} and {kv} as in the first comment at A182760; a(n) is the position of nv. A182766 is the complement of A182765.

Vincenzo Librandi, Table of n, a(n) for n = 1..10000
Index entries for sequences related to Beatty sequences

Cf. A182760, A182765.

[Floor(n*(5-2*Sqrt(2))): n in [1..80]]; // Vincenzo Librandi, Oct 25 2011

Table[Floor[n*(5 - 2*Sqrt[2])], {n, 1, 100}] (* G. C. Greubel, Aug 18 2018 *)

vector(100,n, floor(n*(5-2*sqrt(2)))) \\ G. C. Greubel, Aug 18 2018

a(n) = floor(s*n), where s = 5 - 2*sqrt(2).

A182767 Beatty sequence for 1+e^2.

Original entry on oeis.org

8, 16, 25, 33, 41, 50, 58, 67, 75, 83, 92, 100, 109, 117, 125, 134, 142, 151, 159, 167, 176, 184, 192, 201, 209, 218, 226, 234, 243, 251, 260, 268, 276, 285, 293, 302, 310, 318, 327, 335, 343, 352, 360, 369, 377, 385, 394, 402, 411, 419, 427

Offset: 1

Clark Kimberling, Nov 29 2010

nonn

Let u=e=A001113 and v=1/e=A068985. Jointly rank {j*u} and {k*v} as in the first comment at A182760; a(n) is the position of n*u.

Cf. A182760, A182768.

A182767 := proc(n) floor(n*(1+exp(2))) ; end proc:

a(n)=floor(n*(1+e^2)) = floor(n+n*A072334).

A182768 Beatty sequence for 1+e^(-2).

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 9, 10, 11, 12, 13, 14, 15, 17, 18, 19, 20, 21, 22, 23, 24, 26, 27, 28, 29, 30, 31, 32, 34, 35, 36, 37, 38, 39, 40, 42, 43, 44, 45, 46, 47, 48, 49, 51, 52, 53, 54, 55, 56, 57, 59, 60, 61, 62, 63, 64, 65, 66, 68, 69, 70, 71, 72, 73, 74, 76, 77, 78

Offset: 1

Clark Kimberling, Nov 29 2010

nonn

Let u=e and v=1/e. Jointly rank {ju} and {kv} as in the first comment at A182760; a(n) is the position of nv. A182768 is the complement of A182767.

A182760, A182767.

a(n)=floor(n*(1+e^(-2))).

A182770 Beatty sequence for 3-sqrt(2).

Original entry on oeis.org

1, 3, 4, 6, 7, 9, 11, 12, 14, 15, 17, 19, 20, 22, 23, 25, 26, 28, 30, 31, 33, 34, 36, 38, 39, 41, 42, 44, 45, 47, 49, 50, 52, 53, 55, 57, 58, 60, 61, 63, 65, 66, 68, 69, 71, 72, 74, 76, 77, 79, 80, 82, 84, 85, 87, 88, 90, 91, 93, 95, 96, 98, 99, 101, 103, 104, 106, 107, 109

Offset: 1

Clark Kimberling, Nov 30 2010

nonn

Let u=1+sqrt(2) and v=sqrt(2). Jointly rank {ju} and {kv} as in the first comment at A182760; a(n) is the position of nv. A182770 is the complement of A182769.

Vincenzo Librandi, Table of n, a(n) for n = 1..10000
Index entries for sequences related to Beatty sequences

Cf. A182760, A182769.

[Floor(n*(3-Sqrt(2))): n in [1..80]]; // Vincenzo Librandi, Oct 25 2011

Floor[Range[70](3-Sqrt[2])] (* Harvey P. Dale, Apr 21 2013 *)

vector(100, n, floor(n*(3-sqrt(2)))) \\ G. C. Greubel, Aug 18 2018

a(n) = floor(n*(3-sqrt(2))).

A182772 Beatty sequence for (5-sqrt(3))/2.

Original entry on oeis.org

1, 3, 4, 6, 8, 9, 11, 13, 14, 16, 17, 19, 21, 22, 24, 26, 27, 29, 31, 32, 34, 35, 37, 39, 40, 42, 44, 45, 47, 49, 50, 52, 53, 55, 57, 58, 60, 62, 63, 65, 66, 68, 70, 71, 73, 75, 76, 78, 80, 81, 83, 84, 86, 88, 89, 91, 93, 94, 96, 98, 99, 101, 102, 104

Offset: 1

Clark Kimberling, Nov 30 2010

nonn

Let u=1+sqrt(3) and v=sqrt(3). Jointly rank {ju} and {kv} as in the first comment at A182760; a(n) is the position of nv. A182773 is the complement of A182771.

Vincenzo Librandi, Table of n, a(n) for n = 1..10000

Cf. A182760, A182771.

[Floor(n*(5-Sqrt(3))/2): n in [1..80]]; // Vincenzo Librandi, Oct 25 2011

a(n) = floor(n*(5-sqrt(3))/2).

A379805 Floor of n*(1+sqrt(6))/2.

Original entry on oeis.org

0, 1, 3, 5, 6, 8, 10, 12, 13, 15, 17, 18, 20, 22, 24, 25, 27, 29, 31, 32, 34, 36, 37, 39, 41, 43, 44, 46, 48, 50, 51, 53, 55, 56, 58, 60, 62, 63, 65, 67, 68, 70, 72, 74, 75, 77, 79, 81, 82, 84, 86, 87, 89, 91, 93, 94, 96, 98, 100, 101, 103, 105, 106, 108, 110, 112, 113, 115, 117, 119, 120, 122, 124, 125, 127, 129, 131, 132, 134, 136, 137, 139, 141, 143, 144, 146, 148, 150, 151, 153, 155, 156, 158, 160, 162, 163

Offset: 0

N. J. A. Sloane, Jan 20 2025

nonn

Paolo Xausa, Table of n, a(n) for n = 0..10000

Cf. A000201, A182760, A379800, A379801.

Floor[Range[0, 100]*(1 + Sqrt[6])/2] (* Paolo Xausa, Jan 21 2025 *)

More than the usual number of terms are shown in order to distinguish this from a very similar sequence.

A182771 Beatty sequence for (6+sqrt(3))/3.

Original entry on oeis.org

2, 5, 7, 10, 12, 15, 18, 20, 23, 25, 28, 30, 33, 36, 38, 41, 43, 46, 48, 51, 54, 56, 59, 61, 64, 67, 69, 72, 74, 77, 79, 82, 85, 87, 90, 92, 95, 97, 100, 103, 105, 108, 110, 113, 115, 118, 121, 123, 126, 128, 131, 134, 136, 139, 141, 144, 146, 149

Offset: 1

Clark Kimberling, Nov 30 2010

nonn

Let u=1+sqrt(3) and v=sqrt(3). Jointly rank {j*u} and {k*v} as in the first comment at A182760; a(n) is the position of n*u.

Cf. A182760, A182773.

a(n)=floor(n*(6+sqrt(3))/3).

cp's OEIS Frontend

A190004 A190002/2.

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A182765 Beatty sequence for (6 + sqrt(2))/4.

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A182766 Beatty sequence for 5 - 2*sqrt(2).

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A182767 Beatty sequence for 1+e^2.

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A182768 Beatty sequence for 1+e^(-2).

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A182770 Beatty sequence for 3-sqrt(2).

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Magma

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PARI

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A182772 Beatty sequence for (5-sqrt(3))/2.

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Magma

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A379805 Floor of n*(1+sqrt(6))/2.

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