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A228218 T(n,k)=Number of second differences of arrays of length n+2 of numbers in 0..k.

%I A228218 #8 Jul 23 2025 05:39:38
%S A228218 5,9,15,13,49,31,17,103,199,63,21,177,625,665,127,25,271,1429,3151,
%T A228218 2059,255,29,385,2731,9705,14053,6305,511,33,519,4651,23351,58141,
%U A228218 58975,19171,1023,37,673,7309,47953,176851,320481,242461,58025,2047,41,847,10825
%N A228218 T(n,k)=Number of second differences of arrays of length n+2 of numbers in 0..k.
%C A228218 Table starts
%C A228218 ....5......9......13.......17........21.........25.........29..........33
%C A228218 ...15.....49.....103......177.......271........385........519.........673
%C A228218 ...31....199.....625.....1429......2731.......4651.......7309.......10825
%C A228218 ...63....665....3151.....9705.....23351......47953......88215......149681
%C A228218 ..127...2059...14053....58141....176851.....439927.....951049.....1854553
%C A228218 ..255...6305...58975...320481...1225631....3693505....9399615....21108545
%C A228218 ..511..19171..242461..1688101...8006491...29066311...86929081...224817481
%C A228218 .1023..58025..989527..8717049..50556551..219071473..766106895..2276277137
%C A228218 .2047.175099.4017157.44633821.313882531.1609259287.6537612649.22222129177
%H A228218 R. H. Hardin, <a href="/A228218/b228218.txt">Table of n, a(n) for n = 1..310</a>
%F A228218 Empirical for column k:
%F A228218 k=1: a(n) = 3*a(n-1) -2*a(n-2) for n>3
%F A228218 k=2: a(n) = 5*a(n-1) -6*a(n-2) for n>5
%F A228218 k=3: a(n) = 7*a(n-1) -12*a(n-2) for n>7
%F A228218 k=4: a(n) = 9*a(n-1) -20*a(n-2) for n>9
%F A228218 k=5: a(n) = 11*a(n-1) -30*a(n-2) for n>11
%F A228218 k=6: a(n) = 13*a(n-1) -42*a(n-2) for n>13
%F A228218 k=7: a(n) = 15*a(n-1) -56*a(n-2) for n>15
%F A228218 Empirical for row n:
%F A228218 n=1: a(n) = 4*n + 1
%F A228218 n=2: a(n) = 10*n^2 + 4*n + 1
%F A228218 n=3: a(n) = 20*n^3 + 9*n^2 + 1*n + 1
%F A228218 n=4: a(n) = 35*n^4 + 14*n^3 - 17*n^2 + 30*n + 1
%F A228218 n=5: a(n) = 56*n^5 + 14*n^4 - 108*n^3 + 289*n^2 - 125*n + 1
%F A228218 n=6: a(n) = 84*n^6 - 402*n^4 + 1656*n^3 - 1860*n^2 + 776*n + 1
%F A228218 n=7: a(n) = 120*n^7 - 42*n^6 - 1158*n^5 + 6945*n^4 - 13980*n^3 + 13512*n^2 - 4887*n + 1
%e A228218 Some solutions for n=4 k=4
%e A228218 ..4...-5....3...-3....6...-4...-3...-5...-8....1....5...-2....4...-3...-1...-6
%e A228218 .-6....7....1...-2...-6....1....4....5....6...-5...-5....1....0....0...-3....4
%e A228218 ..2...-3...-1....5....4....2....0...-2...-2....5....7....1....0...-3....3....1
%e A228218 .-2...-1...-2...-1...-6....0...-4....4....1....2...-7....1....4....6...-4...-6
%Y A228218 Row 1 is A004766. A228212 (k=2), A228213 (k=3), A228213 (k=4), A228215 (k=5).
%K A228218 nonn,tabl
%O A228218 1,1
%A A228218 _R. H. Hardin_ Aug 16 2013