A392179 In the spiral discussed in A392177 and A392178, squares not occupied by a knight of either color.

7, 8, 13, 14, 16, 17, 18, 19, 22, 23, 26, 27, 28, 29, 32, 33, 38, 39, 43, 45, 46, 51, 52, 53, 54, 59, 60, 62, 73, 74, 77, 79, 80, 91, 92, 94, 97, 98, 100, 101, 102, 103, 104, 105, 106, 107, 108, 113, 118, 119, 122, 123, 124, 125, 128, 129, 130, 131, 134, 135, 140, 146, 151, 153, 157, 158, 159, 160, 163, 165, 166, 168, 170, 172, 173

Offset

1,1

Comments

A392177, A392178, and this sequence together partition the nonnegative integers.

Links

Michael S. Branicky, Table of n, a(n) for n = 1..50000 (terms 1..3923 from N. J. A. Sloane)

Prog

(Python)
from itertools import count, islice
def square_spiral(): # generator of square spiral coordinates
    i, j, di, dj, L = 0, 0, 1, 0, 1
    yield i, j
    while True:
        for s in range(2):
            for k in range(L):
                i, j = i+di, j+dj
                yield i, j
            di, dj = -dj, di
        L += 1
def agen(): # 0, 1 = Black, Red
    p, g = {0: set(), 1: set()}, {0: square_spiral(), 1: square_spiral()}
    K = {(0, 0), (2, 1), (1, 2), (-1, 2), (-2, 1), (-2, -1), (-1, -2), (1, -2), (2, -1)}
    m, n, occupied = [-1, -1], -1, set()
    while True:
        for turn in (0, 1):
            for k in count(m[turn]+1):
                loc = next(g[turn])
                if all((loc[0]+i, loc[1]+j) not in p[1-turn] for i, j in K):
                    p[turn].add(loc)
                    m[turn] = k
                    occupied.add(k)
                    break
        for n in range(n+1, min(m)+1):
            if n not in occupied:
                yield n
print(list(islice(agen(), 75))) # Michael S. Branicky, Feb 07 2026

Crossrefs

Cf. A392177-A392180.

Sequence in context: A065729 A156605 A343298 * A061905 A308887 A308833

Adjacent sequences: A392176 A392177 A392178 * A392180 A392181 A392182

Keyword

nonn

Author

N. J. A. Sloane, Feb 07 2026

Status

approved