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A284796
Positions of 1's in A284793.
6
1, 4, 7, 9, 12, 16, 19, 22, 25, 28, 31, 33, 36, 40, 43, 45, 48, 52, 55, 58, 61, 63, 66, 70, 73, 76, 79, 81, 84, 88, 91, 94, 97, 100, 103, 105, 108, 112, 115, 117, 120, 124, 127, 130, 133, 136, 139, 141, 144, 148, 151, 153, 156, 160, 163, 166, 169, 171, 174
OFFSET
1,2
COMMENTS
This sequence and A284794 and A284795 form a partition of the positive integers. Conjecture: for n>=1, we have a(n)-3n+3 in {0,1}, 3n+2-A284795(n) in {0,1,2,3}, and 3n-2-A284795(n) in {0,1}.
A284793 = (1,-1,0,1,0,-1,1,-1,1,-1,0,1,0,-1,0,1,0,-1,1,-1,0,1,0,-1, ... ); thus
A284794 = (2,6,8,10,14,...)
A284795 = (3,5,11,13,15,...)
A284796 = (1,4,7,9,12,15,...).
From Michel Dekking, Nov 24 2019: (Start)
Here is a proof of Kimberling's conjecture, i.e., the sequence y defined by y(n) := a(n)-3n+3 takes only values in the alphabet {0,1}. We know that A284793 = 1,-1,0,1,0,-1,... is a morphic sequence(see A284793). Let tau on the alphabet {A,B,C,D} be given by
A -> BC, B->BC, C->ABDC, D->ABDC.
The unique fixed point of tau is x = BCABDCBC... The letter-to letter map pi which gives A284793 = pi(x) is given by
pi(A)=0, pi(B)=1, pi(C)=-1, pi(D)=0.
The return words of B (i.e., the words with prefix B and no other occurrences of B) in x are
a:= BCA, b:= BDC, c:= BC, d:= BDCA.
The morphism tau induces a so-called derivated morphism on the alphabet of return words, which is given by
beta(a) = abc, beta(b) = adb, beta(c) = ab, beta(d) = adbc.
Since B is the unique letter in {A,B,C,D} projecting on the letter 1, the difference sequence Delta*(a(n)) is given by replacing a,b,c,d by their lengths in the fixed point abcadbab... of beta:
a->3, b->3, c->2, d->4.
The difference sequence (Delta (y(n)) is given by
y(n+1)-y(n) = a(n+1)-a(n)-3.
It follows that Delta y only takes the values 0, -1 and 1. Moreover, the 4 words a,b,c,d have projections
pi(BCA)=1,-1,0; pi(BDC)=1,0,-1; pi(BC)=1,-1; pi(BDCA)=1,0,-1,0.
From this we see that 1 and -1 always occur in pairs with 1 first, within the 4 projections of a,b,c, and d. Since y(1)=1, this implies that y itself takes only values in {0,1}.
(End)
LINKS
MATHEMATICA
s = Nest[Flatten[# /. {0 -> {0, 1}, 1 -> {0, 0, 1, 1}}] &, {0}, 7] (* A284775 *)
d = Differences[s] (* A284793 *)
Flatten[Position[d, -1]] (* A284794 *)
Flatten[Position[d, 0]] (* A284795 *)
Flatten[Position[d, 1]] (* A284796 *)
d1/2 (* positions of 0 in A189664 *)
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Clark Kimberling, Apr 14 2017
STATUS
approved