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A228260
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T(n,k)=Number of third differences of arrays of length n+3 of numbers in 0..k
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13
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9, 17, 31, 25, 143, 63, 33, 319, 621, 127, 41, 565, 2511, 2059, 255, 49, 881, 6419, 13933, 6305, 511, 57, 1267, 12947, 53315, 58911, 19171, 1023, 65, 1723, 22727, 141989, 315601, 242461, 58025, 2047, 73, 2249, 36471, 310425, 1161855, 1688101, 989527
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OFFSET
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1,1
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COMMENTS
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Table starts
....9.....17.......25........33.........41..........49..........57...........65
...31....143......319.......565........881........1267........1723.........2249
...63....621.....2511......6419......12947.......22727.......36471........54851
..127...2059....13933.....53315.....141989......310425......596591......1045439
..255...6305....58911....315601....1161855.....3298681.....7795501.....16171769
..511..19171...242461...1688101....8003363....28791007....83538705....206640895
.1023..58025...989527...8717049...50554951...218845881...761638071...2238075697
.2047.175099..4017157..44633821..313882531..1609259287..6537612649..22217937193
.4095.527345.16245775.227363409.1932641711.11658284065.54701587935.211702564321
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LINKS
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FORMULA
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Empirical for column k:
k=1: a(n) = 3*a(n-1) -2*a(n-2) for n>3
k=2: a(n) = 5*a(n-1) -6*a(n-2) for n>5
k=3: a(n) = 7*a(n-1) -12*a(n-2) for n>7
k=4: a(n) = 9*a(n-1) -20*a(n-2) for n>8
k=5: a(n) = 11*a(n-1) -30*a(n-2) for n>10
k=6: a(n) = 13*a(n-1) -42*a(n-2) for n>12
k=7: a(n) = 15*a(n-1) -56*a(n-2) for n>14
Empirical for row n:
n=1: a(n) = 8*n + 1
n=2: a(n) = 35*n^2 + 1*n + 1 for n>1
n=3: a(n) = 112*n^3 - 34*n^2 - 38*n - 13 for n>4
n=4: a(n) = 294*n^4 - 256*n^3 - 489*n^2 + 477*n + 415 for n>8
n=5: a(n) = 672*n^5 - 1152*n^4 - 3340*n^3 + 8908*n^2 + 4108*n - 14279 for n>13
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EXAMPLE
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Some solutions for n=4 k=4
..3...-5....5...-6....3...-7...-8...12....9...13....9...-1....3...-8....7...10
..0....1...-4....7....0....7....3...-7..-12..-13...-3....0...-1...13...-6..-12
.-7....4....6...-7...-6...-7....4...-3...11...13...-5....4...-5..-12....8...12
.10...-8...-1....2...10...-1...-8....8...-8..-15....0...-5....7....8...-6...-9
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CROSSREFS
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KEYWORD
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AUTHOR
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STATUS
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approved
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