A058265 -id:A058265 - cp's OEIS frontend

A158919 Beatty sequence for the tribonacci constant tau (A058265): a(n) = floor(n*tau).

0, 1, 3, 5, 7, 9, 11, 12, 14, 16, 18, 20, 22, 23, 25, 27, 29, 31, 33, 34, 36, 38, 40, 42, 44, 45, 47, 49, 51, 53, 55, 57, 58, 60, 62, 64, 66, 68, 69, 71, 73, 75, 77, 79, 80, 82, 84, 86, 88, 90, 91, 93, 95, 97, 99, 101, 103, 104, 106, 108, 110

Offset: 0

Eric Culver (weux082690(AT)yahoo.com), Mar 30 2009

nonn

Also called the spectrum of tau. Note that A276384 agrees with this sequence for n <= 17160 but disagrees beyond that point. In fact a(17161) = 31564, whereas A276384(17161) = 31563. - N. J. A. Sloane, Sep 03 2016

			a(3) = floor(3*q) = floor(3*1.8392867...) = floor(5.51786...) = 5.

N. J. A. Sloane, Table of n, a(n) for n = 0..20000
A. J. Hildebrand, Junxian Li, Xiaomin Li, Yun Xie, Almost Beatty Partitions, arXiv:1809.08690 [math.NT], 2018.
Wolfdieter Lang, The Tribonacci and ABC Representations of Numbers are Equivalent, arXiv preprint arXiv:1810.09787 [math.NT], 2018.
N. J. A. Sloane, Table of n, a(n) for n = 0..100000
Index entries for sequences related to Beatty sequences

Cf. A058265, A140099 (spectrum of 1+tau), A276384, A277722, A277723.

[Floor(n * (1/3 + 1/3 * (19 - 3 * Sqrt(33))^(1/3) + 1/3 * (19 + 3 * Sqrt(33))^(1/3))) : n in [0..80]]; // Vincenzo Librandi, Oct 28 2018

x := (1/3)*(1+(19+3*sqrt(33))^(1/3)+(19-3*sqrt(33))^(1/3)) ;
[seq(floor(n*x),n=0..200)]; # R. J. Mathar, Sep 11 2011

a[n_] := Floor[n Root[#^3 - #^2 - # - 1&, 1]];
Table[a[n], {n, 0, 100}] (* Jean-François Alcover, Oct 28 2018 *)

a(n) = floor(n*A058265).

A276800 Decimal expansion of t^2, where t is the tribonacci constant A058265.

Original entry on oeis.org

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

Offset: 1

N. J. A. Sloane, Oct 28 2016

The minimal polynomial of this constant is x^3 - 3*x^2 - x - 1, and it is its unique real root. - Amiram Eldar, May 27 2023

			3.38297576790623749412270853645503458694938204374857618201956267723537...

Index entries for algebraic numbers, degree 3

Cf. A058265, A276799, A276801.

A276800L[n_] := RealDigits[(1/3 (1 + (19 - 3 Sqrt[33])^(1/3) + (19 + 3 Sqrt[33])^(1/3)))^2, 10, n][[1]]; A276800L[107] (* JungHwan Min, Nov 06 2016 *)
RealDigits[x /. FindRoot[x^3 - 3*x^2 - x - 1, {x, 3}, WorkingPrecision -> 120]][[1]] (* Amiram Eldar, May 27 2023 *)

polrootsreal(x^3-3*x^2-x-1)[1] \\ Charles R Greathouse IV, Aug 21 2023

A140099 A Beatty sequence: a(n) = [n*(1+t)], where t = tribonacci constant (A058265); complement of A140098.

Original entry on oeis.org

2, 5, 8, 11, 14, 17, 19, 22, 25, 28, 31, 34, 36, 39, 42, 45, 48, 51, 53, 56, 59, 62, 65, 68, 70, 73, 76, 79, 82, 85, 88, 90, 93, 96, 99, 102, 105, 107, 110, 113, 116, 119, 122, 124, 127, 130, 133, 136, 139, 141, 144, 147, 150, 153, 156, 159, 161, 164, 167, 170, 173

Offset: 1

Paul D. Hanna, Jun 01 2008

nonn

Note that A276385 agrees with this sequence for n <= 17160 but disagrees beyond that point. In fact a(17161) = 48725, whereas A276385(17161) = 48724. - N. J. A. Sloane, Sep 03 2016

Also somewhat similar to but different from A109232. - N. J. A. Sloane, Sep 04 2016

			Tribonacci constant: t = 1.839286755214161132551852564653286600...

Harvey P. Dale and N. J. A. Sloane, Table of n, a(n) for n = 1..20000, Aug 29 2016 (First 1000 terms from Harvey P. Dale)
N. J. A. Sloane, Table of n, a(n) for n = 1..100000

Cf. A140098 (complement), A140101, A058265, A109232, A276385.

See also A158919 (Beatty sequence for tribonacci constant tau), A275926 (deviation from A140101).

With[{tc=1/3 (1+Surd[19-3Sqrt[33],3])+1/3 Surd[19+3Sqrt[33],3]},Array[ Floor[ (1+tc)*#]&,70]] (* Harvey P. Dale, Dec 05 2013 *)

{a(n)=local(t=(1+(19+3*sqrt(33))^(1/3)+(19-3*sqrt(33))^(1/3))/3);floor(n*(1+t))}

For n >= 1, a(n) = A158919(n)+n. - N. J. A. Sloane, Sep 04 2016

A276801 Decimal expansion of t^3, where t is the tribonacci constant A058265.

Original entry on oeis.org

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

Offset: 1

N. J. A. Sloane, Oct 28 2016

A cubic integer with minimal polynomial x^3 - 7x^2 + 5x - 1, of which it is the unique real root. - Charles R Greathouse IV, Nov 06 2016

			6.222262523120398626674561101108321187373560789846168428798321316639575...

Index entries for algebraic numbers, degree 3

Cf. A058265, A276800.

RealDigits[x /. FindRoot[x^3 - 7*x^2 + 5*x - 1, {x, 6}, WorkingPrecision -> 120]][[1]] (* Amiram Eldar, May 27 2023 *)

polrootsreal(x^3-7*x^2+5*x-1)[1] \\ Charles R Greathouse IV, Nov 06 2016

1/t + 1/t^2 + 1/t^3 = 1/A058265 + 1/A276800 + 1/A276801 = 1.
From Dimitri Papadopoulos, Nov 07 2023: (Start)
t^3 = (A276800^2 + 1)/2.
t^3 + 1/t^3 = t + 1/t + 4.
t^3 = (1/4)*(t + 1)^2*(t - 1)^2*(t^2 + 1). (End)

A277722 a(n) = floor(n*tau^2) where tau is the tribonacci constant (A058265).

Original entry on oeis.org

0, 3, 6, 10, 13, 16, 20, 23, 27, 30, 33, 37, 40, 43, 47, 50, 54, 57, 60, 64, 67, 71, 74, 77, 81, 84, 87, 91, 94, 98, 101, 104, 108, 111, 115, 118, 121, 125, 128, 131, 135, 138, 142, 145, 148, 152, 155, 158, 162, 165, 169, 172, 175, 179, 182, 186, 189, 192, 196, 199, 202, 206, 209, 213, 216, 219

Offset: 0

N. J. A. Sloane, Oct 30 2016

nonn

JungHwan Min, Table of n, a(n) for n = 0..10000
A. J. Hildebrand, Junxian Li, Xiaomin Li and Yun Xie, Almost Beatty Partitions, arXiv:1809.08690 [math.NT], 2018.

Cf. A058265, A158919, A276800, A277723.

A277722 := proc(n)
    a276800 :=  3.3829757679062374941227085364550345869493820437485761820195626772353718960099402922235933340043661396041006 ;
    floor(n*a276800) ;
end proc:
seq(A277722(n),n=0..100) ; # R. J. Mathar, Nov 02 2016

A277722[n_] := Floor[n (1/3 (1 + (19 - 3 Sqrt[33])^(1/3) + (19 + 3 Sqrt[33])^(1/3)))^2]; Array[A277722, 66, 0] (* JungHwan Min, Nov 06 2016 *)

A276799 a(n) = floor(n*t^2) - A003145(n), where t = 1.8392867... is the tribonacci constant A058265.

Original entry on oeis.org

1, 0, 1, 0, 1, 1, 1, 1, 0, 0, 0, 1, 0, 1, 0, 1, 0, 1, 1, 1, 1, 0, 0, 0, 1, 0, 1, 0, 2, 1, 1, 1, 0, 1, 0, 1, 1, 1, 0, 1, 0, 2, 1, 1, 1, 0, 0, 0, 1, 1, 1, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 1, 1, 1, 0, 0, 0, 0, 1, 0, 1, 0, 1, 1, 1, 1, 0, 0, 0, 1, 1, 1, 0, 1, 0, 1, 1, 1, 1, 0, 0, 0, 1, 0, 1, 0, 1, 0, 1, 1

Offset: 1

N. J. A. Sloane, Oct 28 2016

sign

a(n) is in {-1, 0, 1, 2}, but the first n for which -1 appears is n = 329. - Jeffrey Shallit, Nov 19 2016

N. J. A. Sloane, Table of n, a(n) for n = 1..10000

Cf. A003144, A003145, A003146, A275926, A058265, A276800, A277721, A278352 (positions of -1's).

a(n) = A277722(n) - A003145(n). - R. J. Mathar, Nov 02 2016

A277723 a(n) = floor(n*tau^3) where tau is the tribonacci constant (A058265).

Original entry on oeis.org

0, 6, 12, 18, 24, 31, 37, 43, 49, 56, 62, 68, 74, 80, 87, 93, 99, 105, 112, 118, 124, 130, 136, 143, 149, 155, 161, 168, 174, 180, 186, 192, 199, 205, 211, 217, 224, 230, 236, 242, 248, 255, 261, 267, 273, 280, 286, 292, 298, 304, 311, 317, 323, 329, 336, 342, 348, 354, 360, 367, 373, 379, 385

Offset: 0

N. J. A. Sloane, Oct 30 2016

Jon E. Schoenfield, Table of n, a(n) for n = 0..10000
A. J. Hildebrand, Junxian Li, Xiaomin Li, Yun Xie, Almost Beatty Partitions, arXiv:1809.08690 [math.NT], 2018.

Cf. A058265, A158919, A276801, A277722.

a(n)=n*polrootsreal(x^3-7*x^2+5*x-1)[1]\1 \\ Charles R Greathouse IV, Nov 06 2016

By definition, a(n) = n*tau^3 + O(1). - Charles R Greathouse IV, Nov 06 2016

A140098 A Beatty sequence: a(n) = [n*(1+1/t)], where t = tribonacci constant (A058265); complement of A140099.

Original entry on oeis.org

1, 3, 4, 6, 7, 9, 10, 12, 13, 15, 16, 18, 20, 21, 23, 24, 26, 27, 29, 30, 32, 33, 35, 37, 38, 40, 41, 43, 44, 46, 47, 49, 50, 52, 54, 55, 57, 58, 60, 61, 63, 64, 66, 67, 69, 71, 72, 74, 75, 77, 78, 80, 81, 83, 84, 86, 87, 89, 91, 92, 94, 95, 97, 98, 100, 101, 103, 104, 106

Offset: 1

Paul D. Hanna, Jun 01 2008

nonn

			Tribonacci constant: t = 1.839286755214161132551852564653286600...
1 + 1/t = 1.54368901269207636157085597180174798652520...

N. J. A. Sloane, Table of n, a(n) for n = 1..20000
Eric Weisstein's World of Mathematics, Beatty Sequence.
Index entries for sequences related to Beatty sequences.

Cf. A140099, A140100, A058265.

Floor[Range[100]*(1 + 1/Root[#^3-#^2-#-1 &, 1])] (* Paolo Xausa, Jul 09 2024 *)

{a(n)=local(t=(1+(19+3*sqrt(33))^(1/3)+(19-3*sqrt(33))^(1/3))/3);floor(n*(1+1/t))}

A239502 (Round(q^prime(n)) - 1)/prime(n), where q is the tribonacci constant (A058265).

Original entry on oeis.org

4, 10, 74, 212, 1856, 5618, 53114, 1630932, 5161442, 167427844, 1729192432, 5577731626, 58401766802, 2005139696964, 69737304018266, 228184540445268, 8043367476888770, 86866463049858250, 285815985033409648, 10225367934387562098, 111384745483589787826

Offset: 3

Vladimir Shevelev and Peter J. C. Moses, Mar 20 2014

nonn

For n>=3, round(q^prime(n)) == 1 (mod 2*prime(n)). Proof in Shevelev link. In particular, all terms are even.

			For n=3, q^5 = 21.049..., so a(3) = (21 - 1)/5 = 4.

S. Litsyn and V. Shevelev, Irrational Factors Satisfying the Little Fermat Theorem, International Journal of Number Theory, vol.1, no.4 (2005), 499-512.
V. Shevelev, A property of n-bonacci constant, Seqfan (Mar 23 2014)
Eric Weisstein's World of Mathematics, Tribonacci Constant

Cf. A007619, A007663, A238693, A238697, A238698, A238700, A058265.

A019712 Continued fraction expansion of tribonacci constant A058265.

Original entry on oeis.org

1, 1, 5, 4, 2, 305, 1, 8, 2, 1, 4, 6, 14, 3, 1, 13, 5, 1, 7, 23, 1, 16, 4, 1, 1, 1, 1, 1, 2, 17, 1, 3, 1, 1, 1, 29, 1, 6, 1, 3, 1, 1, 1, 1, 3, 2, 5, 1, 63, 2, 1, 2, 5, 1, 4, 11, 2, 2, 1, 1, 1, 1, 1, 2, 1, 9, 3, 3, 18, 1, 38, 2, 4, 1, 20, 3, 1, 1, 1, 5, 2, 2, 1, 1, 1, 44, 6, 3, 9, 1, 1, 1, 1, 3, 3, 1, 6

Offset: 0

Robert G. Wilson v, Dec 07 2000

The only real root of the equation x^3 - x^2 - x - 1 = 0.

			1.839286755214161132551852564... = 1 + 1/(1 + 1/(5 + 1/(4 + 1/(2 + ...)))). - _Harry J. Smith_, May 30 2009

David Wells, "The Penguin Dictionary of Curious and Interesting Numbers," Revised Edition, Penguin Books, London, England, 1997, page 23.

Harry J. Smith, Table of n, a(n) for n = 0..20000
G. Xiao, Contfrac.
Index entries for continued fractions for constants.

Cf. A058265 (decimal expansion), A319428/A319429 (convergents).

ContinuedFraction[ 1/3 + 1/3*(19 - 3*Sqrt[33])^(1/3) + 1/3*(19 + 3*Sqrt[33])^(1/3), 100]

{ allocatemem(932245000); default(realprecision, 21000); x=contfrac(solve(x=1, 2, x^3 - x^2 - x - 1)); for (n=0, 20000, write("b019712.txt", n, " ", x[n+1])); } \\ Harry J. Smith, May 30 2009

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A158919 Beatty sequence for the tribonacci constant tau (A058265): a(n) = floor(n*tau).

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A276800 Decimal expansion of t^2, where t is the tribonacci constant A058265.

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A140099 A Beatty sequence: a(n) = [n*(1+t)], where t = tribonacci constant (A058265); complement of A140098.

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A276801 Decimal expansion of t^3, where t is the tribonacci constant A058265.

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A277722 a(n) = floor(n*tau^2) where tau is the tribonacci constant (A058265).

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A276799 a(n) = floor(n*t^2) - A003145(n), where t = 1.8392867... is the tribonacci constant A058265.

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A277723 a(n) = floor(n*tau^3) where tau is the tribonacci constant (A058265).

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A140098 A Beatty sequence: a(n) = [n*(1+1/t)], where t = tribonacci constant (A058265); complement of A140099.

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