cp's OEIS Frontend

This sequence and A005843 and A130568 partition the positive integers into sequences with slopes t = 1+sqrt(5), u = 3+sqrt(5), v = 2, where 1/t + 1/u + 1/v = 1. The positions of 1 in A284620 are given by A005843, and of 2, by A130568.

From Michel Dekking, Mar 17 2020: (Start)

This sequence is a generalized Beatty sequence.

It was shown in the Comments of A284620 that A284620 is the letter-to-letter image of the fixed point x = ABCDABCDCD... of the morphism

mu: A->AB, B->CD, C->ABCD, D->CD,

with the letter-to-letter map lambda defined by

lambda: A->0, B->1, C->2, D->1.

Note that A284620(n)=0 iff x(n) = A, where x = ABCDABCDCD... is the fixed point of mu. The return words of A in x are ABCD and ABCDCD. Coding these two return words by their lengths, mu induces a morphism rho on the coded return words given by

rho(4) = 46, rho(4) = 466.

The difference sequence (a(n+1)-a(n)) equals the unique fixed point r = 4646646466... of rho.

The morphism g on the alphabet {a,b} given by

g(a) = ab, g(b) =abb

was introduced in A284620. We see that rho is just an alphabet change of the morphism g.

Let f given by f(b) = ba, f(a) = b be the Fibonacci morphism on the alphabet {b,a} with fixed point x_F = babbababba....

Let x_G = ababbababb... be the fixed point of g. It is well-known (see, e.g., Lemma 12 in "Morphic words..."), that x_G = a x_F.

In general the partial sums of x_F are equal to the generalized Beatty sequence V given by V(n) = p*floor(n*phi) +q*n+r, where p = a-b and q = 2*b-a. See Lemma 8 in the Allouche and Dekking paper. Here we obtain p = 2, q = 2. So a(n) = 2*floor((n-1)*phi) + 2*n - 1, for n>0.

(End)

As a word, A003849 = 01001010010010100..., and replacing each 0010 by 2 gives 0121201012120101201012120101212010...

Warning: "replacing each 0010 by 2" means "replacing each 0010 by 2 from left to right, consecutively". The result is that the word a(8)...a(14)=0010010 in A003849 is replaced by 201, not by 22. - Michel Dekking, Aug 27 2018

A 3-way partition of the positive integers, by positions of 0, 1, 2 in A285677:

A285678: positions of 0; slope t = (4+sqrt(5))/2;

A182761: positions of 1; slope u = (7-sqrt(5))/2;

A285679: positions of 2; slope v = (1+3*sqrt(5))/2;

where 1/t + 1/u + 1/v = 1.

Conjecture: a(n) - a(n-1) is in {2,5} for n>=2.

See A285683 for a proof of this conjecture. - Michel Dekking, Oct 09 2018

a(n) = A285683(n-1) for n>1, see A285683 for a proof. - Michel Dekking, Oct 09 2018

A 3-way partition of the positive integers, by positions of 0, 1, 2 in A285677:

A285678: positions of 0; slope t = (4+sqrt(5))/2;

A182761: positions of 1; slope u = (7-sqrt(5))/2;

A285679: positions of 2; slope v = (1+3*sqrt(5))/2;

where 1/t + 1/u + 1/v = 1. Conjecture: a(n) - a(n-1) is in {2,3,4,5} for n>=2.