



2, 7, 11, 16, 21, 25, 30, 34, 39, 44, 48, 53, 58, 62, 67, 71, 76, 81, 85, 90, 94, 99, 104, 108, 113, 118, 122, 127, 131, 136, 141, 145, 150, 155, 159, 164, 168, 173, 178, 182, 187, 191, 196, 201, 205, 210, 215, 219, 224, 228, 233, 238, 242, 247, 251
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OFFSET

1,1


COMMENTS

Conjecture: 2 < n*r  a(n) < 3 for n >= 1, where r = (7 + sqrt(5))/2.
From Michel Dekking, Feb 12 2021: (Start)
Let T be the transform given by T(0) = 1, T(1) = 011 that defines A287725.
The Fibonacci word A003849 is a fixed point of the morphism sigma: 0>01, 1>0, and therefore also of the morphism sigma^2: 0>010, 1>01.
Now note that T(sigma^2(0)) = T(010) = 10111, T(sigma^2(1)) = T(01) = 1011.
We see from this that the sequence (a(n+1)a(n)) of first differences 5,4,5,5,4,5,4,5,5,4,5,5,4..., of (a(n)) is a sequence on the two letters 4 and 5, and that in fact these two letters occur as the Fibonacci word on the alphabet {5,4}.
From Lemma 8 in the paper by Allouche and Dekking it follows that (a(n)) is the generalized Beatty sequence given by a(n) = floor(n*phi) + 3n  2.
This immediately implies Kimberling's conjecture.
Note that r = (7 + sqrt(5))/2 = phi + 3, where phi is the golden mean. So
n*r  a(n) = n(phi+3)  floor(n*phi) + 3n  2 = n*phi  floor(n*phi)  2,
which lies in (2,3).
These bounds are best possible since the fractional part of n*phi is equidistributed modulo 1.
(End)


LINKS

Clark Kimberling, Table of n, a(n) for n = 1..10000
J.P. Allouche and F. M. Dekking, Generalized Beatty sequences and complementary triples, arXiv:1809.03424 [math.NT], 2018.
J.P. Allouche and F. M. Dekking, Generalized Beatty sequences and complementary triples, Moscow J. Comb. Number Th. 8, 325341, 2019.


FORMULA

a(n) = floor(n*phi) + 3n  2.  Michel Dekking, Feb 12 2021


MATHEMATICA

s = Nest[Flatten[# /. {0 > {0, 1}, 1 > {0}}] &, {0}, 10] (* A003849 *)
w = StringJoin[Map[ToString, s]]
w1 = StringReplace[w, {"0" > "1", "1" > "011"}]
st = ToCharacterCode[w1]  48 (* A287725 *)
Flatten[Position[st, 0]] (* A287726 *)
Flatten[Position[st, 1]] (* A287727 *)


CROSSREFS

Cf. A287725, A287727.
Sequence in context: A140491 A122823 A063289 * A190518 A190750 A049635
Adjacent sequences: A287723 A287724 A287725 * A287727 A287728 A287729


KEYWORD

nonn,easy


AUTHOR

Clark Kimberling, Jun 02 2017


STATUS

approved



