OFFSET
0,3
COMMENTS
I.e. T(i, j) is the smallest n such that w(n) and w(n+1) have the same sign. T(i, j) is zero if i and j have the same sign and T(-i, -j) = T(i, j), so the values tabulated are T(i, -j) = T(-i, j) for 0 <= i, j.
The fact that the T(i, j) and related sequences are well-defined for all i and j can be used to construct dense subrings of the real numbers on the basis of integer arithmetic alone (i.e., without first constructing the real numbers or even the rational numbers). See the first reference.
REFERENCES
R. D. Arthan. An Irrational Construction of R from Z. In Theorem Proving in Higher Order Logics, R. J. Boulton and P.B. Jackson Editors LNCS 2152. Springer Verlag, 2001.
LINKS
EXAMPLE
T(2, -1) = 4 because the generalized Fibonacci sequence 2 -1 1 0 1 1 requires 4 iterations before two consecutive values with the same sign occur.
CROSSREFS
KEYWORD
AUTHOR
Rob Arthan, Sep 18 2001
STATUS
approved