[seqfan] Sum of squared differences of 0..k integers in a circle and periodicity

[Ron Hardin]

Arrange n+1  0..k integers in a cirle.
Square the adjacent differences and add.

What value for the sum can be produced in the most ways?  (pick lowest number for ties)

Obviously the most probable sum keeps rising as n is increased.

At what rate?  About constant rate, giving an almost periodic sequence of first differences, but not quite.

A leap year is added now and then.

More about this behavior I don't know.

Is it a constant rate for large n, and if so is it rational?  Or will the first differences just have leap years added from time to time.

/tmp/dsi
T(n,k)=Most probable value (lowest when ties) of the sum of the squares of differences of adjacent values of a circle of n+1 random 0..k integers

Table starts
..0..2..2..2...2...2...2...2...2...2...2...2...2...2...2...2...2....2....2....2
..2..2..2..2..14..14..14..14..14..14..14..14..14..14..14..14..98...98..182..182
..2..2..6.14..14..26..26..26..26..26..74..74..74.146.146.146.146..194..194..194
..2..6.10.14..26..26..34..46..74..74..86.106.134.134.146.166.206..206..254..274
..2..8.14.16..26..38..46..56..70..86.104.134.134.166.166.214.230..230..304..304
..4..8.16.16..32..40..56..64..88.104.136.136.152.208.208.248.272..304..328..376
..4..8.16.28..40..52..64..88.112.136.160.160.208.232.280.328.328..352..448..472
..4.12.16.28..40..64..76.100.112.160.184.208.232.280.304.352.400..448..520..520
..4.12.24.34..52..70..88.112.136.160.208.250.280.310.370.400.460..520..568..610
..6.14.24.42..54..78.102.126.150.186.222.270.306.354.402.450.510..570..630..690
..6.14.26.42..62..90.114.138.174.210.246.294.342.390.438.498.558..618..690..762
..6.16.30.46..66..90.126.150.186.234.270.318.378.426.486.546.618..690..762..834
..6.18.30.50..74.102.134.170.206.254.302.350.410.470.530.590.674..746..830..914
..8.18.36.56..80.108.140.176.224.272.320.380.440.500.572.644.728..800..896..992
..8.20.36.56..84.116.152.192.236.288.344.404.476.536.620.692.776..872..968.1064
..8.22.40.64..92.124.164.208.256.308.368.436.500.580.656.740.832..928.1028.1136
..8.22.42.68..96.132.172.220.272.328.392.464.536.616.700.788.884..988.1096.1208
.10.24.44.70.102.142.182.234.290.350.418.490.566.650.742.838.938.1046.1162.1282

Empirical for column k (probably not the final word for large n):
k=1: a(n)=a(n-1)+a(n-4)-a(n-5) / resolved with n=1..210
k=2: a(n)=a(n-1)+a(n-3)-a(n-4) for n>12 / resolved with n=1..210
k=3: a(n)=a(n-1)+a(n-4)-a(n-5) for n>44 / resolved with n=1..210
k=4: a(n)=2*a(n-1)-a(n-2) for n>19 / resolved with n=1..210
k=5: a(n)=a(n-1)+a(n-12)-a(n-13) for n>199 / resolved with n=1..410
k=6: a(n)=2*a(n-1)-a(n-2) for n>23 / resolved with n=1..210
k=7: a(n)=a(n-1)+a(n-4)-a(n-5) for n>40 / resolved with n=1..210
k=8: a(n)=a(n-1)+a(n-3)-a(n-4) for n>69 / resolved with n=1..210
k=9: a(n)=a(n-1)+a(n-4)-a(n-5) for n>165 / resolved with n=1..210
k=10: a(n)=2*a(n-1)-a(n-2) for n>50 / resolved with n=1..210
k=11: a(n)=a(n-1)+a(n-12)-a(n-13) for n>124 / resolved with n=1..210
k=12: a(n)=2*a(n-1)-a(n-2) for n>45 / resolved with n=1..210

column 5 first differences with consecutive equals as powers
 12 0 12 0 6 8 0 12 2 8 4 8 6 4 8 4 6 8 4 6^8 4 6^12 4 6^11 4 6^11 4 6^11 4 6^11 4 6^11 4 6^11 4 6^11 4 6^11 4 6^11 4 6^11 4 6^11 4 6^11 4 6^12 4 6^11 4 6^11 4 6^11 4 6^11 4 6^11 4 6^11 4 6^11 4 6^11 4 6^11 4 6^11 4 6^11 4 6^11 4 6^11 4 6^11 4 6^11 4 6^11 4 6^11 4 6^?

Note the 6^12 added halfway through.




 
rhhardin at mindspring.com
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