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

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, 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, 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

Offset: 1

Author

Clark Kimberling, Mar 09 2008

Keywords

• nonn

Comments

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).

Examples

			d(1) - c(c(1)) - 1 =  3 - 1 - 1 = 1;
d(2) - c(c(2)) - 2 =  6 - 2 - 2 = 2;
d(3) - c(c(3)) - 3 =  9 - 5 - 3 = 1;
d(4) - c(c(4)) - 4 = 12 - 7 - 4 = 1.
		

Crossrefs

Cf. A136495, A136496, A138251, A138252.

Formula

A138253(n) = d(n) - c(c(n)) - n, where c(n) = A138251(n), d(n) = A138252(n).