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

Ron Hardin rhhardin at att.net
Sun Apr 14 19:36:51 CEST 2013


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)
> 
> 
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