A064994 -id:A064994 - cp's OEIS frontend

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

Clark Kimberling, Jan 02 2020

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.

Eric Weisstein's World of Mathematics, Beatty Sequence.
Index entries for sequences related to Beatty sequences

Cf. A329825, A329992, A329994 (complement).

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 *)

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.

A064995 A Beatty sequence from Khintchin's constant (A002210).

Original entry on oeis.org

2, 4, 7, 9, 12, 14, 17, 19, 22, 24, 27, 29, 31, 34, 36, 39, 41, 44, 46, 49, 51, 54, 56, 59, 61, 63, 66, 68, 71, 73, 76, 78, 81, 83, 86, 88, 90, 93, 95, 98, 100, 103, 105, 108, 110, 113, 115, 118, 120, 122, 125, 127, 130, 132, 135, 137, 140, 142, 145, 147, 149, 152

Offset: 1

Robert G. Wilson v, Oct 31 2001

nonn

Cf. A002210, A064994.

k = N[Khinchin, 100]; Table[ Floor[ n*(k - 1)/(k - 2) ], {n, 1, 70} ]

cp's OEIS Frontend

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

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