A284796 - OEIS

%I #20 Nov 25 2019 04:59:37

%S 1,4,7,9,12,16,19,22,25,28,31,33,36,40,43,45,48,52,55,58,61,63,66,70,

%T 73,76,79,81,84,88,91,94,97,100,103,105,108,112,115,117,120,124,127,

%U 130,133,136,139,141,144,148,151,153,156,160,163,166,169,171,174

%N Positions of 1's in A284793.

%C 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}.

%C 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

%C A284794 = (2,6,8,10,14,...)

%C A284795 = (3,5,11,13,15,...)

%C A284796 = (1,4,7,9,12,15,...).

%C From _Michel Dekking_, Nov 24 2019: (Start)

%C 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

%C       A -> BC,  B->BC,  C->ABDC,  D->ABDC.

%C The unique fixed point of tau is x = BCABDCBC... The letter-to letter map pi which gives A284793 = pi(x) is given by

%C       pi(A)=0,   pi(B)=1,   pi(C)=-1,   pi(D)=0.

%C The return words of B (i.e., the words with prefix B and no other occurrences of B) in x are

%C       a:= BCA, b:= BDC, c:= BC, d:= BDCA.

%C The morphism tau induces a so-called derivated morphism on the alphabet of return words, which is given by

%C       beta(a) = abc, beta(b) = adb, beta(c) = ab, beta(d) = adbc.

%C 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:

%C       a->3,  b->3,  c->2,  d->4.

%C The difference sequence (Delta (y(n)) is given by

%C       y(n+1)-y(n) = a(n+1)-a(n)-3.

%C It follows that Delta y only takes the values 0, -1 and 1. Moreover, the 4 words a,b,c,d  have projections

%C       pi(BCA)=1,-1,0; pi(BDC)=1,0,-1; pi(BC)=1,-1; pi(BDCA)=1,0,-1,0.

%C 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}.

%C (End)

%H Clark Kimberling, <a href="/A284796/b284796.txt">Table of n, a(n) for n = 1..10000</a>

%t s = Nest[Flatten[# /. {0 -> {0, 1}, 1 -> {0, 0, 1, 1}}] &, {0}, 7] (* A284775 *)

%t d = Differences[s]  (* A284793 *)

%t Flatten[Position[d, -1]] (* A284794 *)

%t Flatten[Position[d, 0]]  (* A284795 *)

%t Flatten[Position[d, 1]]  (* A284796 *)

%t d1/2  (* positions of 0 in A189664 *)

%Y Cf. A284793, A284794, A284795, A189664.

%K nonn,easy

%O 1,2

%A _Clark Kimberling_, Apr 14 2017