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

5, 9, 15, 13, 49, 31, 17, 103, 199, 63, 21, 177, 625, 665, 127, 25, 271, 1429, 3151, 2059, 255, 29, 385, 2731, 9705, 14053, 6305, 511, 33, 519, 4651, 23351, 58141, 58975, 19171, 1023, 37, 673, 7309, 47953, 176851, 320481, 242461, 58025, 2047, 41, 847, 10825

Offset: 1

Author

R. H. Hardin Aug 16 2013

Keywords

• nonn
• tabl

Comments

Table starts

....5......9......13.......17........21.........25.........29..........33

...15.....49.....103......177.......271........385........519.........673

...31....199.....625.....1429......2731.......4651.......7309.......10825

...63....665....3151.....9705.....23351......47953......88215......149681

..127...2059...14053....58141....176851.....439927.....951049.....1854553

..255...6305...58975...320481...1225631....3693505....9399615....21108545

..511..19171..242461..1688101...8006491...29066311...86929081...224817481

.1023..58025..989527..8717049..50556551..219071473..766106895..2276277137

.2047.175099.4017157.44633821.313882531.1609259287.6537612649.22222129177

Examples

			Some solutions for n=4 k=4
..4...-5....3...-3....6...-4...-3...-5...-8....1....5...-2....4...-3...-1...-6
.-6....7....1...-2...-6....1....4....5....6...-5...-5....1....0....0...-3....4
..2...-3...-1....5....4....2....0...-2...-2....5....7....1....0...-3....3....1
.-2...-1...-2...-1...-6....0...-4....4....1....2...-7....1....4....6...-4...-6
		

Links

• R. H. Hardin, Table of n, a(n) for n = 1..310

Crossrefs

Row 1 is A004766. A228212 (k=2), A228213 (k=3), A228213 (k=4), A228215 (k=5).

Formula

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>9

k=5: a(n) = 11*a(n-1) -30*a(n-2) for n>11

k=6: a(n) = 13*a(n-1) -42*a(n-2) for n>13

k=7: a(n) = 15*a(n-1) -56*a(n-2) for n>15

Empirical for row n:

n=1: a(n) = 4*n + 1

n=2: a(n) = 10*n^2 + 4*n + 1

n=3: a(n) = 20*n^3 + 9*n^2 + 1*n + 1

n=4: a(n) = 35*n^4 + 14*n^3 - 17*n^2 + 30*n + 1

n=5: a(n) = 56*n^5 + 14*n^4 - 108*n^3 + 289*n^2 - 125*n + 1

n=6: a(n) = 84*n^6 - 402*n^4 + 1656*n^3 - 1860*n^2 + 776*n + 1

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