%I #12 Nov 12 2024 09:06:14

%S 4,6,10,8,24,20,10,42,88,64,12,64,208,384,136,14,90,426,1242,1606,466,

%T 16,120,728,3030,6856,7138,1012,18,154,1178,6252,22560,43068,31380,

%U 3580,20,192,1744,11524,55372,168506,245860,141272,7864,22,234,2508,19574,123154

%N T(n,k) is the number of length n+1 0..2*k arrays with the sum of the absolute values of adjacent differences equal to n*k.

%H R. H. Hardin, <a href="/A249982/b249982.txt">Table of n, a(n) for n = 1..6839</a>

%F Empirical for row n:

%F n=1: a(n) = 2*n + 2

%F n=2: a(n) = 2*n^2 + 8*n

%F n=3: a(n) = 2*a(n-1) +a(n-2) -4*a(n-3) +a(n-4) +2*a(n-5) -a(n-6); also a polynomial of degree 3 plus a quasipolynomial of degree 1 with period 2

%F n=4: a(n) = (14/3)*n^4 + (56/3)*n^3 + (121/3)*n^2 - (5/3)*n + 2

%F n=5: [linear recurrence of order 10; also a polynomial of degree 5 plus a quasipolynomial of degree 3 with period 2]

%F n=6: a(n) = (1987/180)*n^6 + (2033/30)*n^5 + (2785/18)*n^4 + 226*n^3 - (3017/180)*n^2 + (697/30)*n

%F n=7: [order 14; also a polynomial of degree 7 plus a quasipolynomial of degree 5 with period 2]

%e Table starts:

%e .....4.......6........8........10.........12..........14..........16

%e ....10......24.......42........64.........90.........120.........154

%e ....20......88......208.......426........728........1178........1744

%e ....64.....384.....1242......3030.......6252.......11524.......19574

%e ...136....1606.....6856.....22560......55372......123154......237348

%e ...466....7138....43068....168506.....508902.....1290856.....2886016

%e ..1012...31380...245860...1293326....4598532....14027522....35380112

%e ..3580..141272..1589346...9937894...43752328...152155572...446342246

%e ..7864..635686..9213728..77372824..399919272..1672105528..5529256528

%e .28340.2890884.60568000.604180880.3872197278.18444783546.70948896558

%e Some solutions for n=5 k=4:

%e ..8....5....0....8....3....7....1....0....8....1....1....1....0....0....2....4

%e ..3....1....5....3....8....0....6....3....4....6....2....5....7....2....5....8

%e ..8....6....3....8....0....7....3....2....8....0....6....3....1....8....7....7

%e ..1....4....4....0....2....5....7....8....8....2....0....8....4....6....0....3

%e ..1....8....0....1....4....3....1....1....1....0....5....2....6....0....0....0

%e ..4....3....8....2....1....1....3....4....6....5....1....5....4....4....8....8

%Y Row 1 is A004275(n+2).

%Y Row 2 is A067728.

%K nonn,tabl

%O 1,1

%A _R. H. Hardin_, Nov 10 2014