A391809 A Beatty sequence: floor(n*x) for n >= 1 where x = Sum_{n>=0} 1/2^floor(n*x).

1, 3, 5, 6, 8, 10, 11, 13, 15, 16, 18, 20, 21, 23, 25, 26, 28, 30, 31, 33, 35, 36, 38, 40, 41, 43, 45, 46, 48, 50, 51, 53, 55, 57, 58, 60, 62, 63, 65, 67, 68, 70, 72, 73, 75, 77, 78, 80, 82, 83, 85, 87, 88, 90, 92, 93, 95, 97, 98, 100, 102, 103, 105, 107, 109, 110, 112, 114, 115, 117, 119, 120

Offset

1,2

Comments

A391815 defines the constant x to be x = 1.677419354838709670256625647579633632627970647..., giving x ~ 7554425181395669/4503599627370495 as a rational approximation good to thousands of digits. See A391815 for more details.

Links

Table of n, a(n) for n=1..72.

Eric Weisstein's World of Mathematics, Beatty Sequence.

Example

a(n) = floor(n*x) where
x = 1 + 1/2^1 + 1/2^3 + 1/2^5 + 1/2^6 + 1/2^8 + 1/2^10 + 1/2^11 + 1/2^13 + 1/2^15 + 1/2^16 + 1/2^18 + 1/2^20 + 1/2^21 + ... + 1/2^a(n) + ...
A related sequence is
[0, 1, 1, 2, 2, 3, 4, 4, 5, 5, 6, 7, 7, 8, 8, 9, 10, ..., floor(n/x), ...]
from which the constant x may also be calculated by
x = 1 + 0/2 + 1/2^2 + 1/2^3 + 2/2^4 + 2/2^5 + 3/2^6 + 4/2^7 + 4/2^8 + 5/2^9 + 5/2^10 + 6/2^11 + 7/2^12 + 7/2^13 + ... + floor(n/x)/2^n + ...
The complementary Beatty sequence begins
[2, 4, 7, 9, 12, 14, 17, 19, 22, 24, 27, 29, ..., floor(n*x/(x-1)), ...]
where x = 2 - (1/2^2 + 1/2^4 + 1/2^7 + 1/2^9 + 1/2^12 + 1/2^14 + 1/2^17 + 1/2^19 + 1/2^22 + 1/2^24 + 1/2^27 + ... + 1/2^floor(n*x/(x-1)) + ...).

Prog

(PARI) {a(n) = my(x = 7554425181395669/4503599627370495);
x = suminf(n=0, 1./2^floor(n*x) ) ; floor(n*x)}
for(n=1, 100, print1(a(n), ", "))

Crossrefs

Cf. A391815, A329987 (related Beatty sequence).

Sequence in context: A244644 A047220 A329845 * A329993 A064994 A138235

Adjacent sequences: A391806 A391807 A391808 * A391810 A391811 A391812

Keyword

nonn

Author

Paul D. Hanna, Dec 31 2025

Status

approved