A199697 -id:A199697 - cp's OEIS frontend

A199909 T(n,k)=Number of -k..k arrays x(0..n-1) of n elements with zero sum, and adjacent elements not equal modulo three (with -1 modulo 3 = 2).

1, 1, 2, 1, 4, 6, 1, 4, 12, 8, 1, 6, 24, 24, 14, 1, 8, 42, 72, 82, 32, 1, 8, 60, 152, 256, 232, 56, 1, 10, 84, 256, 804, 1312, 654, 100, 1, 12, 114, 448, 1836, 5016, 5206, 2044, 204, 1, 12, 144, 680, 3196, 12872, 24864, 21208, 6096, 388, 1, 14, 180, 952, 6064, 29864, 77874

Offset: 1

R. H. Hardin Nov 11 2011

Table starts
...1.....1......1.......1........1.........1.........1..........1..........1
...2.....4......4.......6........8.........8........10.........12.........12
...6....12.....24......42.......60........84.......114........144........180
...8....24.....72.....152......256.......448.......680........952.......1384
..14....82....256.....804.....1836......3196......6064......10276......14846
..32...232...1312....5016....12872.....29864.....62776.....114768.....200520
..56...654...5206...24864....77874....216530....518560....1071202....2114394
.100..2044..21208..139148...547604...1699268...4854740...11588992...24551100
.204..6096..97668..814776..3784512..14546928..47329800..125461824..306360336
.388.18564.422052.4509164.25525476.116482068.436295060.1308549932.3582143596

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

R. H. Hardin, Table of n, a(n) for n = 1..871

Column 1 is A199697

Row 2 is A063200(n+2)

Empirical for rows:
T(1,k)=1
T(2,k)=a(k-1)+a(k-3)-a(k-4)
T(3,k)=2*a(k-1)-a(k-2)+a(k-3)-2*a(k-4)+a(k-5)
T(4,k)=a(k-1)+3*a(k-3)-3*a(k-4)-3*a(k-6)+3*a(k-7)+a(k-9)-a(k-10)
T(5,k)=a(k-1)+4*a(k-3)-4*a(k-4)-6*a(k-6)+6*a(k-7)+4*a(k-9)-4*a(k-10)-a(k-12)+a(k-13)
T(6,k)=2*a(k-1)-a(k-2)+4*a(k-3)-8*a(k-4)+4*a(k-5)-6*a(k-6)+12*a(k-7)-6*a(k-8)+4*a(k-9)-8*a(k-10)+4*a(k-11)-a(k-12)+2*a(k-13)-a(k-14)
T(7,k)=a(k-1)+6*a(k-3)-6*a(k-4)-15*a(k-6)+15*a(k-7)+20*a(k-9)-20*a(k-10)-15*a(k-12)+15*a(k-13)+6*a(k-15)-6*a(k-16)-a(k-18)+a(k-19)

A283613 T(n,k) = number of linear arrays of n 1's, n -1's, and k 0's such that no two adjacent elements are equal.

Original entry on oeis.org

1, 1, 2, 6, 6, 2, 2, 12, 30, 38, 24, 6, 2, 18, 74, 174, 248, 212, 100, 20, 2, 24, 138, 480, 1092, 1668, 1700, 1110, 420, 70, 2, 30, 222, 1026, 3228, 7188, 11492, 13140, 10500, 5572, 1764, 252, 2, 36, 326, 1882, 7580, 22274, 48852, 80672, 100044, 91840, 60564, 27132, 7392, 924, 2, 42, 450, 3118, 15324, 56040, 156664, 339720, 574716, 757148, 769356, 591444, 332640, 129096, 30888, 3432, 2, 48, 594, 4804, 27888, 122136, 415576, 1118268, 2403588, 4143116, 5719788, 6281856, 5416488, 3586968, 1760616, 603174, 128700, 12870

Offset: 0

Stefan Hollos, Mar 11 2017

			The table starts with columns k=0...11 and rows n=0...5:
  | 0   1   2    3    4    5     6     7     8    9   10  11
-----------------------------------------------------------
0 | 1   1
1 | 2   6   6    2
2 | 2  12  30   38   24    6
3 | 2  18  74  174  248  212   100    20
4 | 2  24 138  480 1092 1668  1700  1110   420   70
5 | 2  30 222 1026 3228 7188 11492 13140 10500 5572 1764 252
For n=2, k=4 the 24 arrays are:
[-1,0,-1,0,1,0,1,0]  [-1,0,1,0,-1,0,1,0]  [-1,0,1,0,1,0,-1,0]  [1,0,-1,0,-1,0,1,0]
[1,0,-1,0,1,0,-1,0]  [1,0,1,0,-1,0,-1,0]  [0,-1,1,0,-1,0,1,0]  [0,-1,1,0,1,0,-1,0]
[0,-1,0,-1,1,0,1,0]  [0,-1,0,-1,0,1,0,1]  [0,-1,0,1,-1,0,1,0]  [0,-1,0,1,0,-1,1,0]
[0,-1,0,1,0,-1,0,1]  [0,-1,0,1,0,1,-1,0]  [0,-1,0,1,0,1,0,-1]  [0,1,-1,0,-1,0,1,0]
[0,1,-1,0,1,0,-1,0]  [0,1,0,-1,1,0,-1,0]  [0,1,0,-1,0,-1,1,0]  [0,1,0,-1,0,-1,0,1]
[0,1,0,-1,0,1,-1,0]  [0,1,0,-1,0,1,0,-1]  [0,1,0,1,-1,0,-1,0]  [0,1,0,1,0,-1,0,-1]

Cf. A110706, A199697.

nmax=8; Flatten[CoefficientList[Series[CoefficientList[Series[((x + 1)^2*Sqrt[(1 - y)/(1 - (2x + 1)^2*y)] - x - 1)/x, {y, 0, nmax}], y], {x, 0, 2nmax + 1}], x]] (* Indranil Ghosh, Mar 22 2017 *)

G.f.:((x+1)^2*sqrt((1-y)/(1-(2*x+1)^2*y))-x-1)/x.
T(n,0) G.f.: (1+y)/(1-y).
T(n,1) G.f.: (y^2 + 4*y + 1)/(1-y)^2.
T(n,2) G.f.: 2*y*(y^2 + 6*y + 3)/(1-y)^3.
T(n,3) G.f.: 2*y*(2*y^3 + 17*y^2 + 15*y + 1)/(1-y)^4.
T(n,4) G.f.: 4*y^2*(2*y^3 + 23*y^2 + 32*y + 6)/(1-y)^5.
T(n,5) G.f.: 2*y^2*(8*y^4 + 120*y^3 + 243*y^2 + 88*y + 3)/(1-y)^6.
T(n,2*n+1) = binomial(2*n,n).
T(n,2*n) = (n+2)*binomial(2*n,n).
T(n,n) = A110706(n) n > 0.
Sum_{2*n+k = m} T(n,k) = A199697(m).

cp's OEIS Frontend

A199909 T(n,k)=Number of -k..k arrays x(0..n-1) of n elements with zero sum, and adjacent elements not equal modulo three (with -1 modulo 3 = 2).

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