A179008 T(n,k) is the base-2 logarithm of the number of n X k binary arrays with no adjacent elements having the mod 2 sum of their neighbors equal.

1, 1, 1, 2, 3, 2, 1, 1, 1, 1, 1, 1, 3, 1, 1, 1, 3, 1, 1, 3, 1, 2, 1, 2, 5, 2, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 3, 4, 1, 5, 1, 4, 3, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 2, 1, 2, 7, 2, 1, 2, 1, 2, 1, 3, 1, 5, 3, 1, 1, 3, 5, 1, 3, 1, 1, 1, 3, 1, 1, 1, 7, 1, 1, 1, 3, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 3, 2

Offset

1,4

Comments

T(n,k) is apparently the number of bits (not necessarily arbitrarily chosen ones) whose values may be chosen independently, the rest then being determined.

Links

R. H. Hardin, Table of n, a(n) for n = 1..1984

Formula

Empirical: Let x = gcd(k+1,2^k).

T(n,k) = gcd(n+1,k+1) for k or n even;

T(n,k) = gcd(n+1,k+1)-1 for k and n odd with (n+1-x) modulo (2x) = 0;

T(n,k) = gcd(n+1,k+1) otherwise.

Example

Table starts
1 1 2 1 1 1 2 1  1  1  2  1  1  1  2  1  1  1  2  1  1  1  2  1  1 1  2 1  1 1
1 3 1 1 3 1 1 3  1  1  3  1  1  3  1  1  3  1  1  3  1  1  3  1  1 3  1 1  3 1
2 1 3 1 2 1 4 1  2  1  3  1  2  1  4  1  2  1  3  1  2  1  4  1  2 1  3 1  2 1
1 1 1 5 1 1 1 1  5  1  1  1  1  5  1  1  1  1  5  1  1  1  1  5  1 1  1 1  5 1
1 3 2 1 5 1 2 3  1  1  6  1  1  3  2  1  5  1  2  3  1  1  6  1  1 3  2 1  5 1
1 1 1 1 1 7 1 1  1  1  1  1  7  1  1  1  1  1  1  7  1  1  1  1  1 1  7 1  1 1
2 1 4 1 2 1 7 1  2  1  4  1  2  1  8  1  2  1  4  1  2  1  7  1  2 1  4 1  2 1
1 3 1 1 3 1 1 9  1  1  3  1  1  3  1  1  9  1  1  3  1  1  3  1  1 9  1 1  3 1
1 1 2 5 1 1 2 1  9  1  2  1  1  5  2  1  1  1 10  1  1  1  2  5  1 1  2 1  9 1
1 1 1 1 1 1 1 1  1 11  1  1  1  1  1  1  1  1  1  1 11  1  1  1  1 1  1 1  1 1
2 3 3 1 6 1 4 3  2  1 11  1  2  3  4  1  6  1  3  3  2  1 12  1  2 3  3 1  6 1
1 1 1 1 1 1 1 1  1  1  1 13  1  1  1  1  1  1  1  1  1  1  1  1 13 1  1 1  1 1
1 1 2 1 1 7 2 1  1  1  2  1 13  1  2  1  1  1  2  7  1  1  2  1  1 1 14 1  1 1
1 3 1 5 3 1 1 3  5  1  3  1  1 15  1  1  3  1  5  3  1  1  3  5  1 3  1 1 15 1
2 1 4 1 2 1 8 1  2  1  4  1  2  1 15  1  2  1  4  1  2  1  8  1  2 1  4 1  2 1
1 1 1 1 1 1 1 1  1  1  1  1  1  1  1 17  1  1  1  1  1  1  1  1  1 1  1 1  1 1
1 3 2 1 5 1 2 9  1  1  6  1  1  3  2  1 17  1  2  3  1  1  6  1  1 9  2 1  5 1
1 1 1 1 1 1 1 1  1  1  1  1  1  1  1  1  1 19  1  1  1  1  1  1  1 1  1 1  1 1
2 1 3 5 2 1 4 1 10  1  3  1  2  5  4  1  2  1 19  1  2  1  4  5  2 1  3 1 10 1
1 3 1 1 3 7 1 3  1  1  3  1  7  3  1  1  3  1  1 21  1  1  3  1  1 3  7 1  3 1
1 1 2 1 1 1 2 1  1 11  2  1  1  1  2  1  1  1  2  1 21  1  2  1  1 1  2 1  1 1
1 1 1 1 1 1 1 1  1  1  1  1  1  1  1  1  1  1  1  1  1 23  1  1  1 1  1 1  1 1
2 3 4 1 6 1 7 3  2  1 12  1  2  3  8  1  6  1  4  3  2  1 23  1  2 3  4 1  6 1
1 1 1 5 1 1 1 1  5  1  1  1  1  5  1  1  1  1  5  1  1  1  1 25  1 1  1 1  5 1
....
Some solutions for 10 X 10:
  1 1 0 0 1 1 0 1 0 0    0 0 1 0 0 1 1 1 0 0
  1 0 1 0 1 0 0 1 1 1    1 1 1 1 0 1 1 1 0 0
  0 1 1 0 0 0 0 0 1 1    1 1 1 0 0 1 1 1 1 1
  0 0 0 0 1 1 0 1 0 1    1 1 1 0 0 0 0 1 1 1
  1 1 0 1 0 1 1 0 0 1    0 0 0 0 0 0 0 1 1 1
  1 0 0 1 1 1 1 1 1 0    0 1 0 0 1 0 0 0 0 0
  0 0 0 0 1 1 0 0 0 0    1 1 1 0 0 0 0 0 1 0
  1 1 0 1 0 1 0 0 1 1    0 1 0 1 0 0 1 1 1 1
  0 1 1 0 0 1 0 1 0 1    0 0 1 1 1 0 1 1 1 0
  0 1 1 1 1 0 0 1 1 0    1 0 0 1 0 0 1 1 1 0
All solutions for 10 X 9:
  1 0 1 0 1 0 1 0 1    0 0 0 0 0 0 0 0 0
  0 1 0 1 0 1 0 1 0    1 0 1 0 1 0 1 0 1
  1 0 0 0 0 0 0 0 1    0 1 0 1 0 1 0 1 0
  0 1 0 0 0 0 0 1 0    1 0 0 0 0 0 0 0 1
  1 0 0 0 1 0 0 0 1    0 1 0 0 0 0 0 1 0
  0 1 0 0 0 0 0 1 0    1 0 0 0 1 0 0 0 1
  1 0 0 0 0 0 0 0 1    0 1 0 0 0 0 0 1 0
  0 1 0 1 0 1 0 1 0    1 0 0 0 0 0 0 0 1
  1 0 1 0 1 0 1 0 1    0 1 0 1 0 1 0 1 0
  0 0 0 0 0 0 0 0 0    1 0 1 0 1 0 1 0 1
All solutions for 5 X 4:
  0 0 0 0    0 0 0 0
  0 1 0 1    1 0 1 0
  0 0 1 0    0 1 0 0
  0 1 0 1    1 0 1 0
  0 0 0 0    0 0 0 0

Crossrefs

Sequence in context: A083475 A211994 A122402 * A255252 A174985 A008406

Adjacent sequences: A179005 A179006 A179007 * A179009 A179010 A179011

Keyword

nonn,tabl

Author

R. H. Hardin, Jan 03 2011

Status

approved