# [seqfan] Re: Sums of n products of pairs of 0..k integers?

Ron Hardin rhhardin at att.net
Mon Apr 15 14:49:54 CEST 2013

```It seems not to be true above k=31
...
T(n,31)=961*n-222 for n>1
T(n,32)=1024*n-234 for n>2
...
and n>2 until at least k=130.

T(n,32) = 355 1813 2838 3862 4886 5910 6934 7958 8982 10006 11030 12054

first difference

1458 1025 1024 1024 1024 1024 1024 1024 1024 1024 1024

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

----- Original Message ----
> From: Ron Hardin <rhhardin at att.net>
> To: Sequence Fanatics Discussion list <seqfan at list.seqfan.eu>
> Sent: Sun, April 14, 2013 1:36:59 PM
> Subject: [seqfan] Re: Sums of n products of pairs of 0..k integers?
>
> n>2 should read n>1 throughout
>
> 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: Sun,  April 14, 2013 12:36:17 PM
> > Subject: [seqfan] Sums of n products of pairs  of 0..k integers?
> >
> > Question at end
> >
> >  /tmp/dgu
> > T(n,k)=Number of distinct values of the sum  of n products  of two 0..k
> integers
> >
> > Table  starts
> >  ..2..4..7..10..15..19..26..31..37..43..54..60..73.81
> >  ..3..8.16..27..42..59..81.105.134.167.203.241.285...
> >  ..4.12.25..43..67..95.130.169.215.267.324.385.......
> >  ..5.16.34..59..92.131.179.233.296.367.445...........
> >  ..6.20.43..75.117.167.228.297.377.467...............
> >  ..7.24.52..91.142.203.277.361.458...................
> >  ..8.28.61.107.167.239.326.425.......................
> >  ..9.32.70.123.192.275.375...........................
> >  .10.36.79.139.217.311...............................
> >  .11.40.88.155.242...................................
> >
> > Row   1 is A027384
> >
> > Empirical for column k (based on not very  many  points):
> > k=1: a(n) = 1*n + 1
> > k=2: a(n) = 4*n
> >  k=3: a(n) = 9*n -  2
> > k=4: a(n) = 16*n - 5 for n>2
> > k=5:  a(n) = 25*n - 8 for n>2
> > k=6:  a(n) = 36*n - 13 for  n>2
> > k=7: a(n) = 49*n - 17 for n>2
> > k=8: a(n) =  64*n -  23 for n>2
> > k=9: a(n) = 81*n - 28 for n>2
> > k=10: a(n) =  100*n -  33 for n>2
> > k=11: a(n) = 121*n - 39 for n>2
> >  k=12: a(n) = 144*n - 47  for n>2
> > k=13: a(n) = 169*n - 53 for  n>2
> >
> > Apparently after the  second row, each additional  product adds the full k^2
>to
>
>
> > the attained  values.
> >
> > Is this obvious?
> >
> >
> >
> > rhhardin at mindspring.com
> > rhhardin at att.net (either)
> >
> >
> > _______________________________________________
> >
> >  Seqfan  Mailing list - http://list.seqfan.eu/
> >
>
> _______________________________________________
>
> Seqfan Mailing  list - http://list.seqfan.eu/
>
```