A172278 -id:A172278 - cp's OEIS frontend

A172264 a(n) = floor(n*(sqrt(3)-sqrt(2))).

0, 0, 0, 0, 1, 1, 1, 2, 2, 2, 3, 3, 3, 4, 4, 4, 5, 5, 5, 6, 6, 6, 6, 7, 7, 7, 8, 8, 8, 9, 9, 9, 10, 10, 10, 11, 11, 11, 12, 12, 12, 13, 13, 13, 13, 14, 14, 14, 15, 15, 15, 16, 16, 16, 17, 17, 17, 18, 18, 18, 19, 19, 19, 20, 20, 20, 20, 21, 21, 21, 22, 22, 22, 23, 23, 23, 24, 24, 24, 25

Offset: 0

Vincenzo Librandi, Jan 30 2010

Vincenzo Librandi, Table of n, a(n) for n = 0..1000

Cf. A172266-A172270, A172272-A172278.

[Floor(n*(Sqrt(3)-Sqrt(2))): n in [0..80]]; // Vincenzo Librandi, Aug 01 2013

With[{c = Sqrt[3] - Sqrt[2]}, Floor[c Range[0, 100]]] (* Vincenzo Librandi, Aug 01 2013 *)

for(n=0,50, print1(floor(n*(sqrt(3)-sqrt(2))), ", ")) \\ G. C. Greubel, Jul 05 2017

A172272 a(n) = floor(n*(sqrt(11)-sqrt(3))).

Original entry on oeis.org

0, 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, 64, 66, 68, 69, 71, 72, 74, 76, 77, 79, 80, 82, 83, 85, 87, 88, 90, 91, 93, 95, 96, 98, 99, 101, 102, 104, 106, 107

Offset: 0

Vincenzo Librandi, Jan 30 2010

Vincenzo Librandi, Table of n, a(n) for n = 0..1000

Cf. A172264, A172266-A172270, A172273-A172278.

[Floor(n*(Sqrt(11)-Sqrt(3))): n in [0..80]]; // Vincenzo Librandi, Aug 01 2013

With[{c = Sqrt[11] - Sqrt[3]}, Floor[c Range[0, 80]]] (* Vincenzo Librandi, Aug 01 2013 *)

A172266 a(n) = floor(n*(sqrt(5)-sqrt(2))).

Original entry on oeis.org

0, 0, 1, 2, 3, 4, 4, 5, 6, 7, 8, 9, 9, 10, 11, 12, 13, 13, 14, 15, 16, 17, 18, 18, 19, 20, 21, 22, 23, 23, 24, 25, 26, 27, 27, 28, 29, 30, 31, 32, 32, 33, 34, 35, 36, 36, 37, 38, 39, 40, 41, 41, 42, 43, 44, 45, 46, 46, 47, 48, 49, 50, 50, 51, 52, 53, 54, 55, 55, 56, 57, 58, 59, 59

Offset: 0

Vincenzo Librandi, Jan 30 2010

Vincenzo Librandi, Table of n, a(n) for n = 0..1000

Cf. A172264, A172267-A172270, A172272-A172278.

[Floor(n*(Sqrt(5)-Sqrt(2))): n in [0..80]]; // Vincenzo Librandi, Aug 01 2013

With[{c = Sqrt[5] - Sqrt[2]}, Floor[c Range[0, 80]]] (* Vincenzo Librandi, Aug 01 2013 *)

a(n)=(sqrt(5)-sqrt(2))*n\1 \\ Charles R Greathouse IV, Sep 02 2015

A172270 a(n) = floor(n*(sqrt(11)-sqrt(5))).

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 68, 69, 70, 71, 72, 73, 74, 75, 76, 77

Offset: 0

Vincenzo Librandi, Jan 30 2010

Vincenzo Librandi, Table of n, a(n) for n = 0..1000

Cf. A172264, A172266-A172269, A172272-A172278.

[Floor(n*(Sqrt(11)-Sqrt(5))): n in [0..80]]; // Vincenzo Librandi, Aug 01 2013

With[{c=Sqrt[11]-Sqrt[5]},Floor[c*Range[0,80]]] (* Harvey P. Dale, Apr 24 2013 *)

a(n) = floor(n*(sqrt(11)-sqrt(5))) \\ Charles R Greathouse IV, Jul 01 2016

Checked by Michael B. Porter, Jun 16 2010

A316711 Decimal expansion of s:= t/(t - 1), with the tribonacci constant t = A058265.

Original entry on oeis.org

2, 1, 9, 1, 4, 8, 7, 8, 8, 3, 9, 5, 3, 1, 1, 8, 7, 4, 7, 0, 6, 1, 3, 5, 4, 2, 6, 8, 2, 2, 7, 5, 1, 7, 2, 9, 3, 4, 7, 4, 6, 9, 1, 0, 2, 1, 8, 7, 4, 2, 8, 8, 0, 9, 1, 0, 0, 9, 7, 8, 1, 3, 3, 8, 6, 1, 7, 6, 8, 5, 9, 4, 8, 0, 0, 4, 9, 7, 0, 1, 4, 6, 1, 1, 1, 7, 9, 6, 6, 6, 7, 0, 0, 2, 1, 8, 3, 0, 6

Offset: 1

Wolfdieter Lang, Sep 07 2018

Because the tribonacci constant t = A058265 > 1, with Beatty sequence At(n) := floor(n*t), n >= 1 (with At(0) = 0) given in A158919, has the companion sequence Bt := floor(n*s), n >= 1, (with Bt(0) = 0), with 1/t + 1/s = 1, and At and Bt are complementary, disjoint sequences for the positive integers. Note that Bt is not A172278. The first entries n = 0..161 coincide. A172278(162) = 354 but At(193) = A158919(193) = 354, hence A172278 is not complementary together with At. In fact, Bt(162) = 355, which is not a member of At.

s-1 = 1/(t-1) equals the real root of 2*x^3 - 2*x - 1. See the formulas below. - Wolfdieter Lang, Sep 15 2022

			s = 2.191487883953118747061354268227517293474691021874288091009781338617685...

Wolfdieter Lang, The Tribonacci and ABC Representations of Numbers are Equivalent, arXiv preprint arXiv:1810.09787 [math.NT], 2018.
Wikipedia, Beatty Sequence

Cf. A317202, A058265, A158919, A172278, A276801, A357463.

Digits := 120: a := (1/4 + sqrt(33)/36)^(1/3): 1 + a + 1/(3*a): evalf(%)*10^98: ListTools:-Reverse(convert(floor(%), base, 10)); # Peter Luschny, Sep 15 2022

With[{t=x/.Solve[x^3-x^2-x-1==0,x][[1]]},RealDigits[t/(t-1),10,120][[1]]] (* Harvey P. Dale, Sep 12 2021 *)

s = t/(t - 1) with the tribonacci constant t = A058265, the real root of the cubic x^3 - x^2 - x - 1.
s = (1 + (19 + 3*sqrt(33))^(1/3) + (19 - 3*sqrt(33))^(1/3)) / (-2 + (19 + 3*sqrt(33))^(1/3) + (19 - 3*sqrt(33))^(1/3)).
From Wolfdieter Lang, Sep 15 2022: (Start)
s = 1 + ((1 + (1/9)*sqrt(33))/4)^(1/3)+(1/3)*((1 + (1/9)*sqrt(33))/4)^(-1/3).
s = 1 + ((1 + (1/9)*sqrt(33))/4)^(1/3) + ((1 - (1/9)*sqrt(33))/4)^(1/3).
s = 1 + (2/3)*sqrt(3)*cosh((1/3)*arccosh((3/4)*sqrt(3))). (End)
From Dimitri Papadopoulos, Nov 07 2023: (Start)
s = 1 + t^3/(t^3 - 1) = 1 + A276801/(A276801 - 1).
s = 1 + t^2/(t+1). (End)

A172267 a(n) = floor(n*(sqrt(7)-sqrt(5))).

Original entry on oeis.org

0, 0, 0, 1, 1, 2, 2, 2, 3, 3, 4, 4, 4, 5, 5, 6, 6, 6, 7, 7, 8, 8, 9, 9, 9, 10, 10, 11, 11, 11, 12, 12, 13, 13, 13, 14, 14, 15, 15, 15, 16, 16, 17, 17, 18, 18, 18, 19, 19, 20, 20, 20, 21, 21, 22, 22, 22, 23, 23, 24, 24, 24, 25, 25, 26, 26, 27, 27, 27, 28, 28, 29, 29, 29, 30, 30, 31, 31

Offset: 0

Vincenzo Librandi, Jan 30 2010

Vincenzo Librandi, Table of n, a(n) for n = 0..1000

Cf. A172264, A172266-A172270, A172272-A172278.

[Floor(n*(Sqrt(7)-Sqrt(5))): n in [0..80]]; // Vincenzo Librandi, Aug 01 2013

With[{c = Sqrt[7] - Sqrt[5]}, Floor[c Range[0, 80]]] (* Vincenzo Librandi, Aug 01 2013 *)

A172269 a(n) = floor(n*(sqrt(7)-sqrt(2))).

Original entry on oeis.org

0, 1, 2, 3, 4, 6, 7, 8, 9, 11, 12, 13, 14, 16, 17, 18, 19, 20, 22, 23, 24, 25, 27, 28, 29, 30, 32, 33, 34, 35, 36, 38, 39, 40, 41, 43, 44, 45, 46, 48, 49, 50, 51, 52, 54, 55, 56, 57, 59, 60, 61, 62, 64, 65, 66, 67, 68, 70, 71, 72, 73, 75, 76, 77, 78, 80, 81, 82, 83, 84, 86, 87

Offset: 0

Vincenzo Librandi, Jan 30 2010

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Index entries for sequences related to Beatty sequences

Cf. A172264, A172266-A172270, A172272-A172278.

[Floor(n*(Sqrt(7)-Sqrt(2))): n in [0..80]]; // Vincenzo Librandi, Aug 01 2013

With[{c = Sqrt[7] - Sqrt[2]}, Floor[c Range[0, 80]]] (* Vincenzo Librandi, Aug 01 2013 *)

vector(100,n,n--; floor(n*(sqrt(7)-sqrt(2)))) \\ G. C. Greubel, Aug 17 2018

A172273 a(n) = floor(n*(sqrt(11) - sqrt(2))).

Original entry on oeis.org

0, 1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 39, 41, 43, 45, 47, 49, 51, 53, 55, 57, 58, 60, 62, 64, 66, 68, 70, 72, 74, 76, 77, 79, 81, 83, 85, 87, 89, 91, 93, 95, 97, 98, 100, 102, 104, 106, 108, 110, 112, 114, 116, 117, 119, 121, 123, 125

Offset: 0

Vincenzo Librandi, Jan 30 2010

Initially similar to A038124 because sqrt(11)-sqrt(2) = 1.90241122... is close to A065421. - R. J. Mathar, Feb 05 2010

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Index entries for sequences related to Beatty sequences

Cf. A172264, A172266-A172270, A172272-A172278.

[Floor(n*(Sqrt(11)-Sqrt(2))): n in [0..80]]; // Vincenzo Librandi, Aug 01 2013

With[{c = Sqrt[11] - Sqrt[2]}, Floor[c Range[0, 80]]] (* Vincenzo Librandi, Aug 01 2013 *)

vector(100, n, n--; floor(n*(sqrt(11) - sqrt(2)))) \\ G. C. Greubel, Aug 18 2018

A172277 floor(n*(sqrt(13)-sqrt(3))).

Original entry on oeis.org

0, 1, 3, 5, 7, 9, 11, 13, 14, 16, 18, 20, 22, 24, 26, 28, 29, 31, 33, 35, 37, 39, 41, 43, 44, 46, 48, 50, 52, 54, 56, 58, 59, 61, 63, 65, 67, 69, 71, 73, 74, 76, 78, 80, 82, 84, 86, 88, 89, 91, 93, 95, 97, 99, 101, 103, 104, 106, 108, 110, 112, 114, 116, 118, 119, 121, 123

Offset: 0

Vincenzo Librandi, Jan 30 2010

Vincenzo Librandi, Table of n, a(n) for n = 0..1000

Cf. A172264, A172266-A172270, A172272-A172278.

[Floor(n*(Sqrt(13)-Sqrt(3))): n in [0..80]]; // Vincenzo Librandi, Aug 01 2013

A172277:=n->floor(n*(sqrt(13)-sqrt(3))); seq(A172277(k), k=0..100); # Wesley Ivan Hurt, Nov 08 2013

With[{c = Sqrt[13] - Sqrt[3]}, Floor[c Range[0, 80]]] (* Vincenzo Librandi, Aug 01 2013 *)

A276383 Complement of A158919: complementary Beatty sequence to the Beatty sequence defined by the tribonacci constant tau = A058265.

Original entry on oeis.org

2, 4, 6, 8, 10, 13, 15, 17, 19, 21, 24, 26, 28, 30, 32, 35, 37, 39, 41, 43, 46, 48, 50, 52, 54, 56, 59, 61, 63, 65, 67, 70, 72, 74, 76, 78, 81, 83, 85, 87, 89, 92, 94, 96, 98, 100, 102, 105, 107, 109, 111, 113, 116, 118, 120, 122, 124, 127, 129, 131, 133, 135, 138, 140, 142, 144, 146, 149, 151, 153, 155, 157, 159, 162, 164, 166, 168, 170, 173, 175, 177, 179, 181, 184, 186, 188, 190, 192, 195, 197, 199, 201, 203, 205, 208, 210, 212, 214, 216, 219, 221, 223, 225, 227, 230, 232, 234, 236, 238, 241, 243

Offset: 1

N. J. A. Sloane, Sep 02 2016

nonn

This is the Beatty sequence for tau_prime = 2.191487883953118747061354268227517294...,
defined by 1/tau + 1/tau_prime = 1.
Differs from A172278 at n = 162, 209, 256, 303, 324, ...
Note that Beatty sequences do not normally include 0 - see the classic pair A000201, A001950. - N. J. A. Sloane, Oct 19 2018
Note that the tribonacci numbers T = A000073 related to the ternary sequence A080843 lead to the three complementary sequences for the nonnegative integers AT(n) = A278040(n), BT(n) = A278039(n) and CT(n) = A278041(n). - Wolfdieter Lang, Sep 08 2018

			Comments from _Wolfdieter Lang_, Sep 08 2018 (Start):
The complementary sequences A158919 and A276383 begin:
n:       1 2 3 4  5  6  7  8  9 10 11 12 13 14 15 16 17 18 19 20 ...
A158919: 1 3 5 7  9 11 12 14 16 18 20 22 23 25 27 29 31 33 34 36 ...
A276383: 2 4 6 8 10 13 15 17 19 21 24 26 28 30 32 35 37 39 41 43 ...
--------------------------------------------------------------------
The complementary sequences AT, BT and CT begin:
n:  0  1  2  3  4  5  6  7  8  9 10 11 12 13 14 15  16  17  18  19 ...
AT: 1  5  8 12 14 18 21 25 29 32 36 38 42 45 49 52  56  58  62  65 ...
BT: 0  2  4  6  7  9 11 13 15 17 19 20 22 24 26 28  30  31  33  35 ...
CT: 3 10 16 23 27 34 40 47 54 60 67 71 78 84 91 97 104 108 115 121 ...
(End)

R. J. Mathar, Table of n, a(n) for n = 1..1000
Wikipedia, Beatty Sequence

Cf. A000201, A001950, A058265, A158919.
Cf. also A000073, A080843, A276383, A278039, A278040, A278041, A316711.
Similar to but strictly different from A172278.

A276383 := proc(n)
    Tau := (1/3)*(1+(19+3*sqrt(33))^(1/3)+(19-3*sqrt(33))^(1/3));
    taupr := 1/(1-1/Tau) ;
    floor(n*taupr) ;
end proc: # R. J. Mathar, Sep 04 2016
a:=proc(n) local s,t; t:=evalf(solve(x^3-x^2-x-1=0,x),120)[1]; s:=t/(t-1); floor(n*s) end; seq(a(n),n=0..70); # Muniru A Asiru, Oct 16 2018

a(n) = floor(n*tau_prime), with tau_prime = tau/(tau - 1), where tau is the tribonacci constant A058265.

tau_prime = (1 + (19 + 3*sqrt(33))^(1/3) + (19 - 3*sqrt(33))^(1/3)) / (-2 + (19 + 3*sqrt(33))^(1/3) + (19 - 3*sqrt(33))^(1/3)). - Wolfdieter Lang, Sep 08 2018

Edited by N. J. A. Sloane, Oct 19 2018 at the suggestion of Georg Fischer

cp's OEIS Frontend

A172264 a(n) = floor(n*(sqrt(3)-sqrt(2))).

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A172272 a(n) = floor(n*(sqrt(11)-sqrt(3))).

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A172266 a(n) = floor(n*(sqrt(5)-sqrt(2))).

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A172270 a(n) = floor(n*(sqrt(11)-sqrt(5))).

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A316711 Decimal expansion of s:= t/(t - 1), with the tribonacci constant t = A058265.

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A172267 a(n) = floor(n*(sqrt(7)-sqrt(5))).

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A172269 a(n) = floor(n*(sqrt(7)-sqrt(2))).

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A172273 a(n) = floor(n*(sqrt(11) - sqrt(2))).

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