A129700 Number of n-step self-avoiding paths on octant grid starting at octant origin.

1, 1, 2, 3, 8, 14, 36, 70, 177, 372, 942, 2056, 5222, 11736, 29878, 68576, 175038, 408328, 1044533, 2468261, 6326688, 15107015, 38791865, 93432564, 240296399, 583001850, 1501520574, 3665682736, 9452895693, 23201772603, 59899677902

Offset: 1

Author

Bill Blewett, Jun 01 2007

Keywords

• nonn
• more

Comments

Similar to A038373 but with octant grid instead of quadrant. An octant grid is either half of a quadrant grid when divided on the diagonal and including the diagonal grid squares. Its shape is that of a right triangle with a stair step edge on the hypotenuse. Coordinates of squares satisfy x>=0 and y>=0 and x>=y.

Guttmann-Torrie series coefficients c_n^2 for square lattice, with wedge angle Pi/4. - N. J. A. Sloane, Jul 06 2015

Links

• A. J. Guttmann and G. M. Torrie, Critical behavior at an edge for the SAW and Ising model, J. Phys. A 17 (1984), 3539-3552.
• Sean A. Irvine, Java program (github)

Crossrefs

Cf. A038373.

Programs

C

#include 
#include 
#define GRIDSIZE 20
void Recur(int level, int maxlevel, int rgBd[][GRIDSIZE], int i, int j, int rgCt[]) {
  if (i < 0 || j < 0 || i >= GRIDSIZE || j >= GRIDSIZE || level >= maxlevel || j > i || rgBd[i][j] != 0) return;
  rgCt[level] += 1;
  rgBd[i][j] = 1;
  Recur(level + 1, maxlevel, rgBd, i + 1, j, rgCt);
  Recur(level + 1, maxlevel, rgBd, i - 1, j, rgCt);
  Recur(level + 1, maxlevel, rgBd, i, j + 1, rgCt);
  Recur(level + 1, maxlevel, rgBd, i, j - 1, rgCt);
  rgBd[i][j] = 0;
}
int main(int argc, char **argv) {
  int rgBd[GRIDSIZE][GRIDSIZE] = {0};
  int rgCt[GRIDSIZE] = {0};
  int maxlevel = GRIDSIZE;
  if (argc > 1) {
    maxlevel = atoi(argv[1]);
    if (maxlevel < 0 || maxlevel > GRIDSIZE) {
      printf("Bad argument");
      return 0;
    }
  }
  Recur(0, maxlevel, rgBd, 0, 0, rgCt);
  for (int i = 0; i < maxlevel; i++) printf("%2d ", rgCt[i]);
  return 0;
}

Extensions

a(28)-a(31) from Sean A. Irvine, Jul 03 2021