[seqfan] Linear Recurrences T(n, k) version of A189418 rhombus count

Ron Hardin rhhardin at att.net
Sun Apr 24 02:01:28 CEST 2011


http://oeis.org/A189418 the number of rhombi on a nXn grid, if you investigate 
nXk grids, has identical column recurrences

T(n,k)=Number of rhombuses on a (n+1)X(k+1) grid
Table starts
..1..2...3...4...5...6....7....8....9...10...11...12...13...14...15...16...17
..2..6..10..15..20..26...32...39...46...54...62...71...80...90..100..111..122
..3.10..22..36..50..66...82..100..120..142..164..188..212..238..264..292..320
..4.15..36..66..96.130..164..204..248..296..344..396..448..504..560..620..680
..5.20..50..96.151.212..273..344..421..504..587..676..765..860..959.1064.1169
..6.26..66.130.212.312..412..527..650..782..914.1059.1204.1358.1520.1691.1862
..7.32..82.164.273.412..564..736..918.1112.1306.1520.1734.1960.2198.2448.2698
..8.39.100.204.344.527..736..984.1244.1520.1796.2104.2412.2736.3076.3438.3800
..9.46.120.248.421.650..918.1244.1601.1978.2355.2776.3197.3638.4103.4600.5097
.10.54.142.296.504.782.1112.1520.1978.2478.2978.3535.4092.4674.5288.5945.6602

k=1 Empirical: a(n)=2*a(n-1)-a(n-2)
k=2 Empirical: a(n)=2*a(n-1)-2*a(n-3)+a(n-4)
k=3 Empirical: a(n)=2*a(n-1)-2*a(n-3)+a(n-4) for n>11
k=4 Empirical: a(n)=2*a(n-1)-2*a(n-3)+a(n-4) for n>11
k=5 Empirical: a(n)=2*a(n-1)-2*a(n-3)+a(n-4) for n>27
k=6 Empirical: a(n)=2*a(n-1)-2*a(n-3)+a(n-4) for n>27
k=7 Empirical: a(n)=2*a(n-1)-2*a(n-3)+a(n-4) for n>51
k=8 Empirical: a(n)=2*a(n-1)-2*a(n-3)+a(n-4) for n>51
k=9 Empirical: a(n)=2*a(n-1)-2*a(n-3)+a(n-4) for n>83
k=10 Empirical: a(n)=2*a(n-1)-2*a(n-3)+a(n-4) for n>83


if I've done it right, they're all the same (after k=1) except for how far out 
you have to go before the recurrence applies, which seems to be given by 
http://oeis.org/A164897

Is there an obvious explanation?


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



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