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%I #4 Aug 19 2013 08:30:24
%S 9,17,31,25,143,63,33,319,621,127,41,565,2511,2059,255,49,881,6419,
%T 13933,6305,511,57,1267,12947,53315,58911,19171,1023,65,1723,22727,
%U 141989,315601,242461,58025,2047,73,2249,36471,310425,1161855,1688101,989527
%N T(n,k)=Number of third differences of arrays of length n+3 of numbers in 0..k
%C Table starts
%C ....9.....17.......25........33.........41..........49..........57...........65
%C ...31....143......319.......565........881........1267........1723.........2249
%C ...63....621.....2511......6419......12947.......22727.......36471........54851
%C ..127...2059....13933.....53315.....141989......310425......596591......1045439
%C ..255...6305....58911....315601....1161855.....3298681.....7795501.....16171769
%C ..511..19171...242461...1688101....8003363....28791007....83538705....206640895
%C .1023..58025...989527...8717049...50554951...218845881...761638071...2238075697
%C .2047.175099..4017157..44633821..313882531..1609259287..6537612649..22217937193
%C .4095.527345.16245775.227363409.1932641711.11658284065.54701587935.211702564321
%H R. H. Hardin, <a href="/A228260/b228260.txt">Table of n, a(n) for n = 1..160</a>
%F Empirical for column k:
%F k=1: a(n) = 3*a(n-1) -2*a(n-2) for n>3
%F k=2: a(n) = 5*a(n-1) -6*a(n-2) for n>5
%F k=3: a(n) = 7*a(n-1) -12*a(n-2) for n>7
%F k=4: a(n) = 9*a(n-1) -20*a(n-2) for n>8
%F k=5: a(n) = 11*a(n-1) -30*a(n-2) for n>10
%F k=6: a(n) = 13*a(n-1) -42*a(n-2) for n>12
%F k=7: a(n) = 15*a(n-1) -56*a(n-2) for n>14
%F Empirical for row n:
%F n=1: a(n) = 8*n + 1
%F n=2: a(n) = 35*n^2 + 1*n + 1 for n>1
%F n=3: a(n) = 112*n^3 - 34*n^2 - 38*n - 13 for n>4
%F n=4: a(n) = 294*n^4 - 256*n^3 - 489*n^2 + 477*n + 415 for n>8
%F n=5: a(n) = 672*n^5 - 1152*n^4 - 3340*n^3 + 8908*n^2 + 4108*n - 14279 for n>13
%e Some solutions for n=4 k=4
%e ..3...-5....5...-6....3...-7...-8...12....9...13....9...-1....3...-8....7...10
%e ..0....1...-4....7....0....7....3...-7..-12..-13...-3....0...-1...13...-6..-12
%e .-7....4....6...-7...-6...-7....4...-3...11...13...-5....4...-5..-12....8...12
%e .10...-8...-1....2...10...-1...-8....8...-8..-15....0...-5....7....8...-6...-9
%K nonn,tabl
%O 1,1
%A _R. H. Hardin_ Aug 19 2013