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A332789
First differences of the iterated Beatty sequence A007069.
1
1, 3, 2, 2, 2, 1, 3, 1, 3, 2, 1, 3, 1, 3, 2, 2, 2, 1, 3, 2, 2, 2, 1, 3, 1, 3, 2, 2, 2, 1, 3, 2, 2, 2, 1, 3, 1, 3, 2, 1, 3, 1, 3, 2, 2, 2, 1, 3, 1, 3, 2, 1, 3, 1, 3, 2, 2, 2, 1, 3, 2, 2, 2, 1, 3, 1, 3, 2, 1, 3, 1, 3, 2, 2, 2, 1, 3, 1, 3, 2, 1, 3, 1, 3, 2, 2
OFFSET
1,2
COMMENTS
The sequence A007069 is an iterated Beatty sequence, like the Wythoff compound sequence (A003622(n)) = (A000201(A000201(n))), and the sequence (A140868(n)) = (A003151(A003151(n))). This sequence is the sequence of first differences 1, 3, 2, 2, 2, 1, 3, 1, 3, 2, ... of A007069.
(a(n)) is a morphic sequence, i.e., the letter-to-letter image of the fixed point of a morphism.
Let theta on {a,b,c,d} be the morphism given by
theta : a->adc, b->adc, c->ad, d->bc.
Let the letter-to-letter morphism lambda be given by
lambda : a->1, b->2, c->2, d->3.
Then (a(n)) = lambda(x), where x = adcbcadadc... is the unique fixed point of theta.
See the appendix of my paper "Morphic words, Beatty sequences...".
LINKS
Michel Dekking, Morphic words, Beatty sequences and integer images of the Fibonacci language, Theoretical Computer Science 809, 407-417 (2020).
FORMULA
a(n) = A001951(A001951(n+2)+2) - A001951(A001951(n+1)+1).
PROG
(Python)
from math import isqrt
def A332789(n): return isqrt(isqrt(m:=(n+1)**2<<1)**2<<1)-isqrt(isqrt(m-(n<<2)-2)**2<<1) # Chai Wah Wu, Aug 29 2022
CROSSREFS
KEYWORD
nonn
AUTHOR
Michel Dekking, Feb 24 2020
STATUS
approved