|
%I #30 Sep 20 2022 11:09:58
%S 2,5,9,12,15,19,22,26,29,32,36,39,43,46,50,53,56,60,63,67,70,73,77,80,
%T 84,87,90,94,97,101,104,108,111,114,118,121,125,128,131,135,138,142,
%U 145,149,152,155,159,162,166,169,172,176,179,183,186,189,193,196,200
%N Numbers not of the form round(m*sqrt(2)) for any integer m, i.e., complement of A022846.
%C Consider natural numbers A000027 as a triangle 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, etc., then the a(n) indicate rows without a square.
%C Similar to Beatty sequences: where a pair of complementary Beatty sequences are floor(n*c) and floor(n*c/(c-1)) for c an irrational constant > 1, these pairs of complementary sequences are in general round(n*c) and round((n-1/2)*c/(c-1)) for c an irrational constant > 1.
%C This sequence is an inhomogeneous Beatty sequence s(alpha,rho) with slope alpha = 2 + sqrt(2), and intercept rho = -1/2 - sqrt(2)/2. - _Michel Dekking_, Sep 15 2022
%C Let D := 3,4,3,3,4,3,4,3,3,4,3,4,3,4,3,3,4,3,... be the sequence of first differences of (a(n)). It follows from Yasutomi's criterion that D is NOT the fixed point of a morphism. - _Michel Dekking_, Sep 20 2022
%H Harry J. Smith, <a href="/A063957/b063957.txt">Table of n, a(n) for n = 1..1000</a>
%H A. S. Fraenkel, <a href="http://dx.doi.org/10.4153/CJM-1969-002-7">The bracket function and complementary sets of integers</a>, Canadian J. of Math. 21 (1969) 6-27. (Theorem XI)
%H Clark Kimberling, <a href="https://www.emis.de/journals/INTEGERS/papers/q15/q15.Abstract.html">Beatty sequences and trigonometric functions</a>, Integers 16 (2016), #A15.
%H S.-I. Yasutomi, <a href="https://www.researchgate.net/publication/268247171_On_Sturmian_sequences_which_are_invariant_under_some_substitutions">On Sturmian sequences which are invariant under some substitutions</a>, on ResearchGate.
%F a(n) = round((n - 1/2)*(2 + sqrt(2))) = round(n*3.4142...-1.7071...).
%e round(m*sqrt(2)) starts 1,3,4,6,7,8,10,11,13,... so this sequence must start 2,5,9,12,...
%o (PARI) { f=2 + sqrt(2); t=f/2; for (n=1, 1000, write("b063957.txt", n, " ", round(n*f - t)) ) } \\ _Harry J. Smith_, Sep 03 2009
%Y Cf. A001951, A001952, A007064, A022846.
%K nonn
%O 1,1
%A _Henry Bottomley_, Sep 04 2001
|
