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A268361
Lexicographically least sequence of a certain form that avoids additive squares.
1
1, 2, 1, 3, 1, 2, 1, 5, 1, 2, 1, 3, 1, 2, 1, 8, 1, 2, 1, 3, 1, 2, 1, 5, 1, 2, 1, 3, 1, 2, 1, 13, 1, 2, 1, 3, 1, 2, 1, 5, 1, 2, 1, 3, 1, 2, 1, 8, 1, 2, 1, 3, 1, 2, 1, 5, 1, 2, 1, 3, 1, 2, 1, 21, 1, 2, 1, 3, 1, 2, 1, 5, 1, 2, 1, 3, 1, 2, 1, 8, 1, 2, 1, 3, 1
OFFSET
1,2
COMMENTS
An "additive square" consists of two consecutive blocks of the same length, with the same sum. This sequence a is the lexicographically least sequence over the natural numbers, having no additive square, which is of the form a(i) = b(A001511(i)) for some sequence b.
LINKS
Chen Fei Du, Hamoon Mousavi, Luke Schaeffer, and Jeffrey Shallit, Decision Algorithms for Fibonacci-Automatic Words, with Applications to Pattern Avoidance, arXiv:1406.0670 [cs.FL], 2014.
FORMULA
a(n) = A000045(A001511(n)+1).
G.f.: Sum_{j>=1} A000045(j+1)*x^(2^(j-1))/(1-x^(2^j)). - Robert Israel, Feb 03 2016
Multiplicative with a(2^e) = Fibonacci(2 + e), a(p^e) = 1 for odd prime p. - Andrew Howroyd, Aug 02 2018
Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = 3. - Amiram Eldar, Nov 30 2022
MAPLE
f:= n -> combinat:-fibonacci(padic:-ordp(2*n, 2)+1):
map(f, [$1..100]); # Robert Israel, Feb 03 2016
MATHEMATICA
a[n_] := Fibonacci[IntegerExponent[2n, 2] + 1];
Array[a, 100] (* Jean-François Alcover, Aug 29 2018 *)
PROG
(PARI) a(n)=fibonacci(2 + valuation(n, 2)) \\ Andrew Howroyd, Aug 02 2018
CROSSREFS
Sequence in context: A059127 A319494 A105609 * A330569 A340785 A355758
KEYWORD
nonn,mult
AUTHOR
Jeffrey Shallit, Feb 03 2016
STATUS
approved