A138251 -id:A138251 - cp's OEIS frontend

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

Clark Kimberling, Mar 09 2008

nonn

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).

			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.

Cf. A136495, A136496, A138251, A138252.

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

A138252 Beatty sequence of the number t satisfying 1/s + 1/t = 1, where s is the positive root of x^3 - x^2 - 1.

Original entry on oeis.org

3, 6, 9, 12, 15, 18, 22, 25, 28, 31, 34, 37, 40, 44, 47, 50, 53, 56, 59, 62, 66, 69, 72, 75, 78, 81, 84, 88, 91, 94, 97, 100, 103, 107, 110, 113, 116, 119, 122, 125, 129, 132, 135, 138, 141, 144, 147, 151, 154, 157, 160, 163, 166, 169, 173, 176, 179, 182, 185, 188

Offset: 1

Clark Kimberling, Mar 09 2008

nonn

Complement of A138251.

First differs from A110117 at 34th term.

Cf. A110117, A138252, A138253.

a(n)=Floor(t*n).

Typo in data corrected by D. S. McNeil, Aug 17 2010

cp's OEIS Frontend

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

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