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

Author

N. J. A. Sloane, Sep 02 2016

Keywords

• nonn

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

Examples

			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)
		

Links

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

Crossrefs

Cf. A000201, A001950, A058265, A158919.

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

Similar to but strictly different from A172278.

Programs

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

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

Extensions

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