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A330118
Beatty sequence for 1+x+x^2, where 1/(1+x) + 1/(1+x+x^2) = 1.
2
2, 4, 6, 9, 11, 13, 16, 18, 20, 23, 25, 27, 30, 32, 34, 37, 39, 41, 44, 46, 48, 51, 53, 55, 58, 60, 62, 65, 67, 69, 72, 74, 76, 79, 81, 83, 86, 88, 90, 92, 95, 97, 99, 102, 104, 106, 109, 111, 113, 116, 118, 120, 123, 125, 127, 130, 132, 134, 137, 139, 141
OFFSET
1,1
COMMENTS
Let x be the positive solution of 1/(1+x) + 1/(1+x+x^2) = 1. Then (floor(n*(1+x)) and (floor(n*(1+x+x^2))) 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.
FORMULA
a(n) = floor(n (1+x))), where x = 0.7548776662... is the constant in A075778.
MATHEMATICA
r = x /. FindRoot[1/(1 + x) + 1/(1 + x + x^2) == 1, {x, 1, 2}, WorkingPrecision -> 200]
RealDigits[r] (* A075778 *)
Table[Floor[n*(1 + r)], {n, 1, 250}] (* A330117 *)
Table[Floor[n*(1 + r + r^2)], {n, 1, 250}] (* A330118 *)
Plot[1/(1 + x) + 1/(1 + x + x^2) - 1, {x, 0, 2}]
CROSSREFS
Cf. A329825, A075778, A330117 (complement).
Sequence in context: A187339 A292649 A061785 * A047292 A189930 A184627
KEYWORD
nonn,easy
AUTHOR
Clark Kimberling, Jan 04 2020
STATUS
approved