This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
# Using graphillion
from graphillion import GraphSet
def make_CnXPk(n, k):
grids = []
for i in range(1, k + 1):
for j in range(1, n):
grids.append((i + (j - 1) * k, i + j * k))
grids.append((i + (n - 1) * k, i))
for i in range(1, k * n, k):
for j in range(1, k):
grids.append((i + j - 1, i + j))
return grids
def A(start, goal, n, k):
universe = make_CnXPk(n, k)
GraphSet.set_universe(universe)
paths = GraphSet.paths(start, goal)
return paths.len()
def A338963(n):
m = n * n
s = 0
for i in range(1, m):
for j in range(i + 1, m + 1):
s += A(i, j, n, n)
return s
print([A338963(n) for n in range(3, 7)])
[3,30] cat [128*3^(n-2)-(21*n+57)*2^(n-2): n in [3..30]]; // Vincenzo Librandi, Apr 27 2014
Join[{3,30},LinearRecurrence[{7,-16,12},{144,588,2160},30]] (* Harvey P. Dale, Apr 26 2014 *)
CoefficientList[Series[3 (1 + 3 x - 6 x^2 + 8 x^3 - 4 x^4)/((1 - 3 x) (1 - 2 x)^2), {x, 0, 30}], x] (* Vincenzo Librandi, Apr 27 2014 *)
# Using graphillion
from graphillion import GraphSet
def make_CnXPk(n, k):
grids = []
for i in range(1, k + 1):
for j in range(1, n):
grids.append((i + (j - 1) * k, i + j * k))
grids.append((i + (n - 1) * k, i))
for i in range(1, k * n, k):
for j in range(1, k):
grids.append((i + j - 1, i + j))
return grids
def A(start, goal, n, k):
universe = make_CnXPk(n, k)
GraphSet.set_universe(universe)
paths = GraphSet.paths(start, goal, is_hamilton=True)
return paths.len()
def B(n, k):
m = k * n
s = 0
for i in range(1, m):
for j in range(i + 1, m + 1):
s += A(i, j, n, k)
return s
def A003689(n):
return B(3, n)
print([A003689(n) for n in range(1, 21)]) # Seiichi Manyama, Dec 18 2020
With n=2 there are four sectors: North-Western, North-Eastern, South-Western, South Eastern. Eight nonintersecting (self-avoiding) rook paths joining opposite poles exist: NorthPole NW SW SouthPole NorthPole NW SW SE SouthPole NorthPole NW NE SE SouthPole NorthPole NW NE SE SW SouthPole NorthPole NE SE SouthPole NorthPole NE SE SW SouthPole NorthPole NE NW SW SouthPole NorthPole NE NW SW SE SouthPole So a(2)=8.
#include// GCC -O3 // a(7) in ~1.5 hours char grid[8][8]; long long SIZE; long long calc_ways(long long x, long long y) { if (grid[x][y]) return 0; grid[x][y] = 1; long long n = calc_ways( x==0? SIZE-1 : x-1, y); // try West if (SIZE>2) n+= calc_ways( x==SIZE-1? 0 : x+1, y); // East if (y>0) n+= calc_ways(x,y-1); // North if (y==SIZE-1) n++; else n+= calc_ways(x,y+1); // South grid[x][y] = 0; return n; } int main(int argc, char **argv) { for (SIZE=1; SIZE<7; ++SIZE) { memset(grid, 0, sizeof(grid)); printf("%llu, ",calc_ways(0,0)*SIZE); } printf("\n "); for (SIZE=3; SIZE<9; ++SIZE) { unsigned long long r; memset(grid, 0, sizeof(grid)); grid[0][0]=1; grid[0][1]=1; r = calc_ways(0,2)*SIZE; if (SIZE>6) printf("."); r += calc_ways(1,1)*SIZE*2; if (SIZE>6) printf("."); grid[0][1]=0; grid[1][0]=1; r += calc_ways(1,1)*SIZE*2; if (SIZE>6) printf("."); r += calc_ways(2,0)*SIZE*2; printf("%llu, ", r); } }
# Using graphillion
from graphillion import GraphSet
def make_CnXPk(n, k):
grids = []
for i in range(1, k + 1):
for j in range(1, n):
grids.append((i + (j - 1) * k, i + j * k))
grids.append((i + (n - 1) * k, i))
for i in range(1, k * n, k):
for j in range(1, k):
grids.append((i + j - 1, i + j))
return grids
def A(start, goal, n, k):
universe = make_CnXPk(n, k)
GraphSet.set_universe(universe)
paths = GraphSet.paths(start, goal, is_hamilton=True)
return paths.len()
def B(n, k):
m = k * n
s = 0
for i in range(1, m):
for j in range(i + 1, m + 1):
s += A(i, j, n, k)
return s
def A338297(n):
return B(6, n)
print([A338297(n) for n in range(1, 11)])
Comments