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

[Ron Hardin]

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)