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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.

User: Peter Selinger

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Peter Selinger has authored 2 sequences.

A375299 Number of longest winning paths in n X n Hex.

Original entry on oeis.org

1, 3, 1, 4, 23, 51, 20, 115, 5568, 12, 3521, 40, 1058, 2104, 668, 7540, 1298, 83648, 16631833, 70630
Offset: 1

Views

Author

Peter Selinger, Aug 13 2024

Keywords

Comments

A winning path is a set of cells connecting the top edge to the bottom edge, minimal with respect to inclusion.

Examples

			For n = 4, the longest winning path has length 8, and the a(4) = 4 winning paths of length 8 are
   ========     ========     ========     ========
    X . . .      . X . .      X . . .      . X . .
     X . X X      X . X X      X . X X      X . X X
      X X . X      X X . X      X X . X      X X . X
       . . . X      . . . X      . . X .      . . X .
       ========     ========     ========     ========

Links

Crossrefs

Length of longest path is A375298.
Cf. A357516.

A375298 Length of the longest winning path in n X n Hex.

Original entry on oeis.org

1, 2, 5, 8, 11, 16, 23, 30, 37, 47, 57, 69, 81, 94, 109, 124, 140, 157, 175, 195, 215, 236, 259, 282, 306, 331, 357, 385, 413, 442, 473, 504, 536, 569, 603, 639, 675, 712, 751, 790, 830, 871, 913, 957, 1001, 1046, 1093, 1140, 1188, 1237, 1287, 1339, 1391, 1444, 1499, 1554, 1610, 1667, 1725, 1785
Offset: 1

Views

Author

Peter Selinger, Aug 12 2024

Keywords

Comments

A winning path is a set of cells connecting the top edge to the bottom edge, minimal with respect to inclusion.

Examples

			The longest winning path for 10 X 10 Hex has length 47.
====================
 X . . . . . . . . .
  X X X X X X X X X X
   . . . . . . . . . X
    . X X X . X X X . X
     X . . X X . . X . X
      X X . . . X X . X .
       . X . X X . . . X X
        X . X . . X X . . X
         X . X X X . X X X .
          X . . . . . . . . .
          ====================

Links

Crossrefs

Number of solutions is A375299.
Cf. A357234.

Programs

  • Python
    def a(n):
  if n == 1:
    return 1
  elif n == 2:
    return 2
  elif n == 3:
    return 5
  elif n == 4:
    return 8
  elif n == 9:
    return 37
  elif n % 8 == 3:
    return (2*n**2 - n + 1) // 4 - 1
  else:
    return (2*n**2 - n + 1) // 4

Formula

Provably for n >= 10: a(n) = n^2/2 - n/4 - 3/4 if n ≡ 3 (mod 8), and a(n) = floor(n^2/2 - n/4 + 1/4) otherwise.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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