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EXAMPLE
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The triangle begins:
1,
1, 3;
1, 4, 7;
1, 7, 11, 15;
1, 11, 21, 26, 31;
1, 18, 39, 51, 57, 63;
1, 29, 71, 99, 113, 120, 127;
1, 47, 131, 191, 223, 239, 247, 255;
1, 76, 241, 367, 439, 475, 493, 502, 511;
1, 123, 443, 708, 863, 943, 983, 1003, 1013, 1023;
An m-path cover of a graph G is a covering of the vertices of G by paths of length at most m.
The length of a path is the number of vertices in it. Let C_4 have vertices {1, 2, 3, 4} then c_3(4) = 11 because C_4 has 11 3-path covers: 1, 2, 3, 4; 12, 3, 4; 23, 1, 4; 34, 1, 2; 14, 2, 3; 12, 34; 14, 23; 123, 4; 234, 1; 143, 2; 214, 3. Here an m-path and its reverse are considered to be the same m-path.
If we include the starting conditions c_m(i) = 2^i - 1, 1 <= i <= m, we get the following square:
1, 1, 1, 1, 1, 1, 1, 1, 1, 1 ...
1, 3, 3, 3, 3, 3, 3, 3, 3, 3 ...
1, 4, 7, 7, 7, 7, 7, 7, 7, 7 ...
1, 7, 11, 15, 15, 15, 15, 15, 15, 15 ...
1, 11, 21, 26, 31, 31, 31, 31, 31, 31 ...
1, 18, 39, 51, 57, 63, 63, 63, 63, 63 ...
1, 29, 71, 99, 113, 120, 127, 127, 127, 127 ...
1, 47, 131, 191, 223, 239, 247, 255, 255, 255 ...
1, 76, 241, 367, 439, 475, 493, 502, 511, 511 ...
1, 123, 443, 708, 863, 943, 983, 1003, 1013, 1023 ...
...
Column m of the above square is formed from an m-anacci recurrence.
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