[seqfan] Re: Orthogonsl (-x,y) vectors, was (no subject)

Ron Hardin rhhardin at att.net
Mon May 13 18:50:46 CEST 2013

```"Entries appear to be zero only when the vector lengths of (-x,y) vectors is a
multiple of x+y."

Should be "NOT a multiple," which of course the table avoids.  Skipped row
entries are apparently all zero, which is what the question is about.

Let me also add a subject line!

rhhardin at mindspring.com
rhhardin at att.net (either)

----- Original Message ----
> From: Ron Hardin <rhhardin at att.net>
> To: seqfan at list.seqfan.eu
> Sent: Mon, May 13, 2013 12:03:58 PM
> Subject: [seqfan] (no subject)
>
> Question at end
>
> /tmp/dhi
> T(n,k)=Number of pairs of orthogonal (-x,y)  vectors of length k*(x+y), where
>x/y
>
> is the n-th rational number <= 1,  ordered first by y and then x, e.g. 1/1,
>1/2,
>
> 1/3, 2/3, 1/4, 3/4  .
>
> Table  starts
> ...4......48..........640.............8960...............129024
> ...9.....390........20160..........1106424.............62606544
> ..20....3192.......652784........134432480..........28244153600
> ..60...48720.....41801760......36951200000.......33792065269760
> ..45...26010.....21594300......15593341800.......11432293516320
> .420.8168160.106322711040.1496349265582080.22046445329510891520
>
> row  1 (two column vectors)
> All solutions for k=1
> .-1.-1...-1..1...-1..1....1..1
> .-1..1....1..1...-1.-1...-1..1
>
> row  2 (two column vectors)
> All solutions for k=1
> .-1.-1...-1..2...-1..2...-1..2....2..2...-1..2...-1.-1....2..2...-1..2
> .-1..2...-1.-1....2.-1....2..2...-1..2....2..2...-1.-1...-1..2...-1..2
> .-1.-1...-1.-1....2..2....2.-1....2.-1...-1..2...-1..2...-1..2....2..2
>
> row  3 (two column vectors)
> Some solutions for k=1
> .-1..3...-1..3....3..3...-1..3...-1..3...-1..3...-1..3...-1.-1....3..3...-1..3
> ..3..3...-1..3...-1..3....3..3...-1..3....3.-1...-1..3...-1.-1...-1..3....3.-1
> .-1..3....3..3...-1..3....3.-1....3.-1....3..3...-1..3...-1.-1....3.-1....3..3
> ..3.-1....3.-1....3.-1...-1..3....3..3...-1..3....3..3...-1..3....3.-1....3.-1
>
> row  4 (two column vectors)
> Some solutions for k=1
> .-2..3...-2..3...-2..3....3..3...-2.-2...-2..3...-2.-2...-2.-2...-2..3...-2..3
> .-2..3....3.-2....3..3...-2..3...-2..3...-2..3...-2..3...-2..3....3.-2....3.-2
> ..3..3....3..3...-2..3...-2..3...-2.-2...-2..3...-2.-2...-2.-2....3..3...-2..3
> .-2..3....3..3....3..3....3..3...-2..3....3..3...-2.-2....3.-2....3..3....3..3
> ..3..3....3.-2...-2..3....3.-2...-2.-2....3..3....3.-2...-2.-2...-2..3....3..3
>
> row  5 (two column vectors)
> Some solutions for k=1
> .-1..4...-1..4....4..4...-1..4...-1..4...-1.-1...-1..4....4..4...-1..4...-1..4
> ..4..4...-1..4...-1..4....4.-1....4..4...-1.-1....4.-1...-1..4....4.-1....4..4
> ..4.-1...-1..4...-1..4....4.-1...-1..4...-1.-1....4..4...-1..4....4..4...-1..4
> ..4.-1....4..4...-1..4...-1..4...-1..4...-1..4...-1..4....4.-1....4.-1...-1..4
> .-1..4...-1..4...-1..4....4..4...-1..4...-1.-1....4.-1....4.-1...-1..4....4.-1
>
> row  6 (two column vectors)
> Some solutions for k=1
> .-3..4...-3..4...-3..4...-3..4....4..4...-3..4....4..4...-3.-3...-3.-3....4..4
> .-3..4....4..4...-3.-3....4..4...-3..4....4.-3...-3..4...-3.-3...-3.-3....4..4
> .-3..4....4.-3...-3.-3....4..4....4..4....4..4....4..4...-3..4...-3..4...-3..4
> ..4..4....4..4...-3..4...-3..4...-3..4....4.-3...-3..4...-3..4...-3..4....4.-3
> ..4.-3....4.-3...-3.-3....4.-3....4.-3...-3..4....4..4....4.-3...-3.-3...-3..4
> ..4..4...-3..4...-3.-3....4..4....4..4....4..4....4.-3...-3.-3...-3.-3....4..4
> ..4..4....4..4...-3..4...-3..4....4.-3....4..4...-3..4...-3.-3...-3..4....4.-3
>
> row  7 (two column vectors)
> Some solutions for k=1
> .-1..5...-1..5...-1..5...-1..5...-1..5...-1..5...-1..5...-1..5...-1..5...-1..5
> .-1..5...-1..5...-1.-1...-1..5....5.-1...-1..5...-1..5....5..5....5.-1....5.-1
> .-1..5...-1..5...-1.-1....5.-1....5.-1...-1..5....5..5...-1..5...-1..5....5..5
> ..5..5...-1..5...-1.-1....5..5...-1..5....5.-1...-1..5...-1..5...-1..5...-1..5
> .-1..5....5..5...-1.-1...-1..5...-1..5....5.-1...-1..5....5.-1....5..5....5.-1
> .-1..5....5.-1...-1.-1...-1..5....5..5....5..5....5.-1...-1..5....5.-1....5.-1
>
> Entries  appear to be zero only when the vector lengths of (-x,y) vectors is a

> multiple of x+y.
>
> Is this easy to prove?  Does a formula for  column 1 fall out of it?
>
> Column 1 starts
>  4 9 20 60 45 420 102 378  1120 3024 231 22176 520 2376 8160 23760 63360 164736
>
> 1161 21780 185328  1235520 2570
>
> Row 1 is http://oeis.org/A098402
>
>
> rhhardin at mindspring.com
> rhhardin at att.net (either)
>
>
> _______________________________________________
>
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>
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