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A287775 Positions of 0 in A287772; complement of A050140 (conjectured and proved). 4
2, 3, 6, 7, 9, 10, 13, 14, 17, 18, 20, 21, 24, 25, 27, 28, 31, 32, 35, 36, 38, 39, 42, 43, 46, 47, 49, 50, 53, 54, 56, 57, 60, 61, 64, 65, 67, 68, 71, 72, 74, 75, 78, 79, 82, 83, 85, 86, 89, 90, 93, 94, 96, 97, 100, 101, 103, 104, 107, 108, 111, 112, 114 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,1

COMMENTS

-1 < n*r - a(n) < 1 for n >= 1, where r = (5 + sqrt(5))/4.

From Michel Dekking, Dec 28 2017: (Start)

Let (d(n)) be the sequence of first differences: d(n)=a(n+1)-a(n).

CLAIM: d(n) = A108103(n+1) for n=1,2,….

Proof: As a word A287772 = 100110010011001100100110010011... obtained by substituting 0->1, 1->00 in the Fibonacci word F=0100101001001010010100...

This implies that A287772 is a concatenation of 00’s separated by 1’s and 11’s. Moreover, a 0110 occurs iff 1001 occurs in F, and a 010 occurs iff 101 occurs in F. Note also that occurrence of a 00 in A287772 yields a d(n)=1 (and so every other letter in d is a 1), occurrence of a 010 yields a d(n)=2, and occurrence of a 0110 yields a d(n)=3. Since the 1001’s and 101’s occur in 1F according to F itself with 1 prepended (see A001468 and A282162), we must have d(n)=A108103(n+1). (End)

LINKS

Clark Kimberling, Table of n, a(n) for n = 1..10000

MATHEMATICA

s = Nest[Flatten[# /. {0 -> {0, 1}, 1 -> {0}}] &, {0}, 10] (* A003849 *)

w = StringJoin[Map[ToString, s]]

w1 = StringReplace[w, {"0" -> "1", "1" -> "00"}]

st = ToCharacterCode[w1] - 48    (* A287772 *)

Flatten[Position[st, 0]]  (* A287775 *)

Flatten[Position[st, 1]]  (* A050140 conjectured *)

CROSSREFS

Cf. A050140, A287772, A108103.

Sequence in context: A191215 A190847 A201734 * A283208 A259726 A259587

Adjacent sequences:  A287772 A287773 A287774 * A287776 A287777 A287778

KEYWORD

nonn,easy

AUTHOR

Clark Kimberling, Jun 03 2017

STATUS

approved

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Last modified October 22 07:48 EDT 2019. Contains 328315 sequences. (Running on oeis4.)