[seqfan] Re: A possible characterization of A125121

David Wilson davidwwilson at comcast.net
Sun Nov 17 19:51:07 CET 2013

```William: You can tile the integers with 35 by placing the first bit at
positions 3k for integer k.

Allan: Indeed, you were tiling the integers while I was tiling the
nonnegative integers.

Now that the water has been sufficiently muddied, perhaps I can provide some
clarification:

In this thread, we have discussed the following conditions on tileable
numbers:

R: right shift allowed (as opposed to left shift only).
F: Bit reversal (flip) allowed.
N vs Z: Numbers tile the nonnegative integers vs the integers.

This leads to the following possible variants of tileable numbers:

Z-tileable numbers: No such numbers, right shift required to tile negative
integers.
FZ-tileable numbers: No such numbers, right shift required to tile negative
integers.
RZ-tileable numbers: Distinct from sturdy numbers A125121 (69 is in A125121
but not Z-tileable).
RFZ-tileable numbers: Strict superset of Z-tileables (69 is RZ-tileable but
not Z-tileable).
N-tileable numbers: Strict superset of LN-tileable numbers. Numbers of the
form 2^k * LN-tileable number.
FN-tileable numbers: Strict superset of LRN-tileable numbers. Numbers of the
form 2^k * LRN-tileable number.
LN-tileable numbers: All odd. Strict superset of A064896 (51 is LN-tileable,
but not in A064896).
LFN-tileable numbers. All odd. Strict superset of LN-tileable numbers (11 is
LRN-tileable but not LN-tileable).

This leads to 6 distinct variant sequences of tileable numbers, none of
which are presently in the OEIS.
If others are amenable, I will reserve 6 consecutive A-numbers for these
sequences and try to fill them in.

The RZ-tileable numbers were Allan's original idea on this thread.
Allan's earlier conjecture that RZ-tileable = A125121 is incorrect, see
above.

The LN-tileable numbers were my misinterpretation of Allan's idea.
My earlier statement LN-tileable = A064896 is also incorrect, see above.

> -----Original Message-----
> From: SeqFan [mailto:seqfan-bounces at list.seqfan.eu] On Behalf Of William
> Keith
> Sent: Sunday, November 17, 2013 12:02 PM
> To: Sequence Fanatics Discussion list
> Subject: [seqfan] Re: A possible characterization of A125121
>
> On Sat, Nov 16, 2013 at 6:22 PM, Allan Wechsler <acwacw at gmail.com>
> wrote:
>
> > I think I am having a terminology difference with David Wilson. I
> > intended my "tilings" to cover all the integers, not just the
> > non-negative ones, and thus right shifts are not only allowed, they are
> necessary.
> >
> > This also explains our difference of opinion about 35.  I agree 100011
> > can't tile just the non-negative integers, but it *can* tile all the
> > integers, as I thought I showed in a previous message.
> >
>
> More importantly, don't you also need flips, which don't correspond to
> multiplications?
>
> 100011
> 0100011 would collide
>
> 100011
> 00100011
> -------------
> 10101111 but now what do you do about those two single 0s?  You can cover
> them if you take
>
> 110001
> 00110001
> 0000100011
> 000000100011
> --------------------
> 111111111111 and now you can tile the integers.
>
> But reversal is not a simple arithmetic operation.
>
> William Keith
>
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```