A329993 Beatty sequence for x^2, where 1/x^2 + 1/2^x = 1.

1, 3, 5, 6, 8, 10, 11, 13, 15, 16, 18, 20, 21, 23, 25, 26, 28, 30, 32, 33, 35, 37, 38, 40, 42, 43, 45, 47, 48, 50, 52, 53, 55, 57, 58, 60, 62, 64, 65, 67, 69, 70, 72, 74, 75, 77, 79, 80, 82, 84, 85, 87, 89, 91, 92, 94, 96, 97, 99, 101, 102, 104, 106, 107

Offset

1,2

Comments

Let x be the solution of 1/x^2 + 1/2^x = 1. Then (floor(n x^2)) and (floor(n 2^x)) are a pair of Beatty sequences; i.e., every positive integer is in exactly one of the sequences. See the Guide to related sequences at A329825.

Links

Table of n, a(n) for n=1..64.

Eric Weisstein's World of Mathematics, Beatty Sequence.

Index entries for sequences related to Beatty sequences

Formula

a(n) = floor(n*x^2), where x = 1.29819... is the constant in A329992; a(n) first differs from A064994(n) at n=89.

Mathematica

r = x /. FindRoot[1/x^2 + 1/2^x == 1, {x, 1, 10}, WorkingPrecision -> 120]
RealDigits[r][[1]] (* A329992 *)
Table[Floor[n*r^2], {n, 1, 250}]  (* A329993 *)
Table[Floor[n*2^r], {n, 1, 250}]  (* A329994 *)

Crossrefs

Cf. A329825, A329992, A329994 (complement).

Sequence in context: A244644 A047220 A329845 * A064994 A138235 A329833

Adjacent sequences: A329990 A329991 A329992 * A329994 A329995 A329996

Keyword

nonn,easy

Author

Clark Kimberling, Jan 02 2020

Status

approved