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

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

1,1

Comments

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

Links

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

Wikipedia, Beatty Sequence

Formula

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

Example

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)

Maple

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

Crossrefs

Cf. A000201, A001950, A058265, A158919.

Cf. also A000073, A080843, A276383, A278039, A278040, A278041, A316711.

Similar to but strictly different from A172278.

Sequence in context: A182766 A329828 A172278 * A094390 A186289 A332687

Adjacent sequences: A276380 A276381 A276382 * A276384 A276385 A276386

Keyword

nonn

Author

N. J. A. Sloane, Sep 02 2016

Extensions

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

Status

approved