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A064034 2-dimensional table T(i, j) defined for any integers i and j, read by antidiagonals in the southeast quadrant. T(i, j) gives the "Fibonacci depth" of (i, j): form the Fibonacci sequence starting with i, j: w(0) = i, w(1) = j, w(n) = w(n-1) + w(n-2). It can be shown that for all but finitely many n, the w(n) have the same sign, i.e., are all positive, all negative or all zero. T(i, j), is the smallest number of iterations required to find out which of these cases holds.

%I A064034 #12 Jul 20 2017 23:30:50
%S A064034 0,1,2,1,3,2,1,1,4,2,1,1,3,2,2,1,1,1,5,2,2,1,1,1,3,4,2,2,1,1,1,1,3,2,
%T A064034 2,2,1,1,1,1,3,6,2,2,2,1,1,1,1,1,3,4,2,2,2,1,1,1,1,1,3,5,2,2,2,2,1,1,
%U A064034 1,1,1,1,3,4,2,2,2,2,1,1,1,1,1,1,3,3,4,2,2,2,2,1,1,1,1,1,1,1,3,7,2,2,2,2,2
%N A064034 2-dimensional table T(i, j) defined for any integers i and j, read by antidiagonals in the southeast quadrant. T(i, j) gives the "Fibonacci depth" of (i, j): form the Fibonacci sequence starting with i, j: w(0) = i, w(1) = j, w(n) = w(n-1) + w(n-2). It can be shown that for all but finitely many n, the w(n) have the same sign, i.e., are all positive, all negative or all zero. T(i, j), is the smallest number of iterations required to find out which of these cases holds.
%C A064034 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.
%C A064034 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.
%D A064034 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.
%H A064034 R. D. Arthan <a href="http://www.lemma-one.com/papers/papers.html">An Irrational Construction of R from Z</a>
%e A064034 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.
%Y A064034 Cf. A000045.
%K A064034 nonn,tabl,nice
%O A064034 0,3
%A A064034 _Rob Arthan_, Sep 18 2001