# Nice new sequences, and some problems.

Antti Karttunen karttu at megabaud.fi
Sun Feb 4 00:21:06 CET 2001

```About the question I posed in my previous mail:

> %C A059662 Question: If A059661 could be extended infinitely, would all the natural numbers eventually appear here once, or would some still be missing (1,6,7,?)

Of course the bit 1 will never be flipped again (because
it's already included in the first term, 2), and maybe the
question makes more sense as: "will every natural number > 1
eventually occur in A059662, and furthermore, will there
be any further terms of the form 2^n - 1 in A059661, after 7 and 31 ?"
(I guess the statistics are against that.)

But here's another question, from the sequence A059648 I submitted
yesterday:

ID Number: A059648
Sequence:  0,0,1,0,0,0,0,1,0,1,0,0,1,0,1,0,0,0,0,1,0,0,0,0,1,0,1,0,0,0,
0,1,0,0,0,0,1,0,1,0,0,1,0,1,0,0,0,0,1,0,1,0,0,1,0,1,0,0,0,0,
1,0,0,0,0,1,0,1,0,0,1,0,1,0,0,0,0,1,0,1,0,0,1,0,1,0,0,0,0,1,
0,0,0,0,1,0,1,0,0,0,0,1,0,0,0
Name:      a(n) = [[(k^2)*n]-(k*[k*n])], where k = sqrt(2) and [] is the floor
function.
Comments:  The values of (floor((k^2)*j)-(k*(floor(k*j)))) for j=0..20, with
k=sqrt(2), are 0, 0.585786, 1.171572, 0.343144, 0.928930, 0.100502,
0.68629, 1.27207, 0.44365, 1.02943, 0.20100, 0.78679, 1.37258,
0.54415, 1.12993, 0.30151, 0.88729, 0.05886, 0.64465, 1.23044,
0.40201
Problem: for which values of k are such sequences as A059651, A059652
always positive? Conjecture: k must be quadratic, i.e. have periodic
continued fraction. However, that alone is not sufficient condition,
as A059652 demonstrates. Furthermore, I have not even checked whether
even this sequence stays always positive (but it looks like it would, as
is the case for any k=sqrt(D), where D is a non-square integer).
Maple:     Digits := 89; floor_diffs_floored(sqrt(2),120);
floor_diffs_floored := proc(k,upto_n) local j;
[seq(floor(floor((k^2)*j)-(k*(floor(k*j)))),j=0..upto_n)];
end;

Does anybody want to compute more of these sequences for
various irrational constants k (using as large precision as possible,
to avoid any rounding errors "to the wrong side of the integer fence"),
from which we might find more examples from which to home to the
exact nature of those constants on which these sequences stay
non-negative? (or find a nice proof directly, without all this
"empirical science")
Similar sequences for famous transcedentals like Pi or e might also
be interesting. (They seem to contain negative terms).

The only certain case I know is k = tau = (1+sqrt(5))/2, for
which I know that the resulting sequence is A000004, the zero sequence.

Terveisin,

Antti Karttunen

```