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A138253 Beatty discrepancy of the complementary equation b(n) = a(a(n)) + n.

%I A138253 #11 Jan 31 2025 01:28:45
%S A138253 1,2,1,1,0,1,1,1,0,1,0,1,0,1,2,1,1,0,1,0,2,1,1,0,1,0,0,0,1,1,1,1,0,2,
%T A138253 1,1,0,1,0,0,1,1,0,1,1,0,1,1,1,1,1,0,1,0,1,0,1,0,0,1,1,2,1,1,0,1,0,1,
%U A138253 0,1,0,1,1,0,2,1,1,0,1,0,1,1,1,0,1,0,0,2,1,2,1,1,0,1,1,1,0,1,0,1,0,1,2,1,1
%N A138253 Beatty discrepancy of the complementary equation b(n) = a(a(n)) + n.
%C A138253 Suppose that (a(n)) and (b(n)) are complementary sequences that satisfy a complementary equation b(n) = f(a(n), n) and that the limits r = lim_{n->inf} a(n)/n and s = lim_{n->inf} b(n)/n exist and are both in the open interval (0,1). Let c(n) = floor(a(n)) and d(n) = floor(b(n)), so that (c(n)) and (d(n)) are a pair of Beatty sequences. Define e(n) = d(n) - f(c(n), n). The sequence (e(n)) is here introduced as the Beatty discrepancy of the complementary equation b(n) = f(a(n), n). In the case at hand, (e(n)) measures the closeness of the pair (A136495, A136496) to the Beatty pair (A138251, A138252).
%F A138253 A138253(n) = d(n) - c(c(n)) - n, where c(n) = A138251(n), d(n) = A138252(n).
%e A138253 d(1) - c(c(1)) - 1 =  3 - 1 - 1 = 1;
%e A138253 d(2) - c(c(2)) - 2 =  6 - 2 - 2 = 2;
%e A138253 d(3) - c(c(3)) - 3 =  9 - 5 - 3 = 1;
%e A138253 d(4) - c(c(4)) - 4 = 12 - 7 - 4 = 1.
%Y A138253 Cf. A136495, A136496, A138251, A138252.
%K A138253 nonn
%O A138253 1,2
%A A138253 _Clark Kimberling_, Mar 09 2008