Kishore Rajesh - cp's OEIS frontend

User: Kishore Rajesh

Kishore Rajesh has authored 2 sequences.

A362368 Number of binary strings of length n which are losing configurations in the palindrome game.

0, 0, 2, 0, 4, 0, 18, 4, 56, 12, 156, 80, 568, 424, 1856, 2080, 6548, 8524, 22430, 33840, 80672, 132704, 292428, 510892, 1079282, 1955388, 3990564, 7453012, 14928434, 28406028, 56125298, 108156096, 212297776

Offset: 0

Kishore Rajesh, Apr 17 2023

The palindrome game is a game where players take turns removing a nonempty palindrome from a binary string. The player who crosses off the last palindrome wins. The nonempty palindrome can be removed from anywhere in the current string.

			For n = 2, 10 and 01 are losing strings since the first player has to cross out either the first or second character, leaving the second player with a string of length 1, which is always a palindrome. 00 and 11 are not losing strings since the first player can cross out the entire string and win.
For n = 4, the four losing positions are 0011, 0101, 1010, 1100.
For n = 5, 00101 is winning since the first player may put the second in a losing position by removing 010 from the center or by removing either of the first two 0's.

from functools import lru_cache
from itertools import product
def ispal(s): return s == s[::-1]
def m(s): yield from (s[:i]+s[j:] for i in range(len(s)) for j in range(i+1, len(s)+1) if ispal(s[i:j]))
@lru_cache(maxsize=None)
def L(s): return all(not L(t) for t in m(s))
def a(n): return 2*sum(1 for p in product("01", repeat=n-1) if L("0"+"".join(p))) if n else 0
print([a(n) for n in range(16)]) # Michael S. Branicky, May 23 2023

a(21)-a(29) from Michael S. Branicky, May 23 2023
a(30)-a(31) from Michael S. Branicky, May 26 2023
a(32) from Michael S. Branicky, May 30 2023

A361697 The least y-value of the lower left corner of an n X n box with x-value n such that no edge of the box overlaps with a previous box, given that the first box has its lower left corner at (1,1).

Original entry on oeis.org

1, 2, 3, 4, 2, 6, 3, 8, 4, 7, 9, 12, 1, 11, 3, 16, 2, 13, 4, 17, 5, 20, 6, 24, 8, 14, 1, 25, 7, 21, 3, 32, 10, 22, 9, 31, 2, 23, 11, 37, 4, 26, 12, 42, 13, 29, 5, 48, 16, 33, 6, 40, 17, 28, 8, 53, 14, 36, 1, 51, 15, 35, 3, 64, 7, 46, 18, 56, 10, 44, 9, 67

Offset: 1

Kishore Rajesh, Mar 20 2023

nonn

The sequence is defined by the following method: Draw a 1 X 1 box with lower left corner at (1,1) on the coordinate grid. Then, for all integers n greater than 1, a(n) is the smallest positive integer value such that an n X n box with lower left coordinate (n, a(n)) has no edges that overlap with other boxes.

A029744 appears to be the terms for which a(n)=n in this sequence.

			The following shows the positioning of the first 5 boxes:
            __ __ __ __
           |   __ __ __|__ __
         __|__|__      |     |
        |  |  |  |     |     |
      __|__|__|__|__ __|     |
     |  |__|__|__|           |
   __|__ __|  |__ __ __ __ __|
  |__|

Neal Gersh Tolunsky, Table of n, a(n) for n = 1..10000
Rémy Sigrist, PARI program
Kishore Rajesh, Illustration of first 10 terms of the sequence.

Cf. A029744, A131117.

Cf. A289523 (analogous sequence with circles), A361742 (variant allowing overlapping edges).

PARI
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See Links section.
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cp's OEIS Frontend

User: Kishore Rajesh

A362368 Number of binary strings of length n which are losing configurations in the palindrome game.

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