



1, 5, 6, 10, 14, 15, 19, 20, 24, 28, 29, 33, 37, 38, 42, 43, 47, 51, 52, 56, 57, 61, 65, 66, 70, 74, 75, 79, 80, 84, 88, 89, 93, 97, 98, 102, 103, 107, 111, 112, 116, 117, 121, 125, 126, 130, 134, 135, 139, 140, 144, 148, 149, 153, 154, 158, 162, 163, 167
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OFFSET

1,2


COMMENTS

Conjecture: 0 < n*r  a(n) < 3 for n >= 1, where r = (1 + 3*sqrt(5))/2. [Corrected by Clark Kimberling, Aug 19 2019]
From Michel Dekking, Aug 21 2019: (Start)
Proof of Maiga's conjecture: let T denote the morphism {0>1, 1>000}.
The Fibonacci word xF:=A003849 is fixed point of the morphism 0>01, 1>0 and therefore xF is a concatenation of the two words v=01 and w=0, where these words occur as the Fibonacci word itself.
Now note that
T(v) = T(01) = 1000, T(w) = T(0) = 1.
We see from this that the sequence of first differences of A287665, Delta A287665 = 4,1,4,4,1,4,1,4,4,1,..., is a sequence on the letters 4 and 1, and that in fact these two letters occur as the Fibonacci word on the alphabet {4,1}.
Since A001468 (starting from n=1) is the Fibonacci word on the alphabet {2,1}, Maiga's formula follows.
Proof of Kimberling's conjecture.
It follows from the result above by Lemma 8 in the AlloucheDekking paper that A287665 is a generalized Beatty sequence
a(n) = 3*floor(n*phi)  2*n.
So if r = (1 + 3*sqrt(5))/2 = 3*phi  2, then
n*r  a(n) = n*(3*phi2)  [3*(n*phi{n*phi})]2*n = 3*{n*phi}, where {} denotes fractional part.
It follows that 0 < n*r  a(n) < 3. Moreover, these bounds are tight, since the sequence ({n*phi}) is equidistributed on (0,1).
(End)


LINKS

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


FORMULA

a(n) = Sum_{k=0..n1} [3*A001468(k)2] (conjectured).  Jon Maiga, Dec 30 2018
a(n) = 3*floor(n*phi)  2*n.  Michel Dekking, Aug 21 2019


MATHEMATICA

s = Nest[Flatten[# /. {0 > {0, 1}, 1 > {0}}] &, {0}, 10] (* A003849 *)
w = StringJoin[Map[ToString, s]]
w1 = StringReplace[w, {"0" > "1", "1" > "000"}]
st = ToCharacterCode[w1]  48 (* A287663 *)
Flatten[Position[st, 0]] (* A287664 *)
Flatten[Position[st, 1]] (* A287665 *)


CROSSREFS

Cf. A287663, A287664, A001468.
Sequence in context: A166563 A005847 A109758 * A015820 A096728 A329271
Adjacent sequences: A287662 A287663 A287664 * A287666 A287667 A287668


KEYWORD

nonn,easy


AUTHOR

Clark Kimberling, Jun 02 2017


STATUS

approved



