A382355 A version of the Josephus problem: a(n) is the surviving integer under the skip-eliminate-skip version of the elimination process.

1, 1, 1, 4, 3, 6, 3, 6, 9, 3, 6, 9, 12, 1, 4, 7, 10, 13, 16, 19, 1, 4, 7, 10, 13, 16, 19, 22, 25, 28, 31, 3, 6, 9, 12, 15, 18, 21, 24, 27, 30, 33, 36, 39, 42, 45, 1, 4, 7, 10, 13, 16, 19, 22, 25, 28, 31, 34, 37, 40, 43, 46, 49, 52, 55, 58, 61, 64, 67, 70, 3, 6

Offset

1,4

Comments

This variation of the Josephus problem is related to under-down-under card dealing.

Links

Table of n, a(n) for n=1..72.

Eric Huang, Tanya Khovanova, Timur Kilybayev, Ryan Li, Brandon Ni, Leone Seidel, Samarth Sharma, Nathan Sheffield, Vivek Varanasi, Alice Yin, Boya Yun, and William Zelevinsky, Card Dealing Math, arXiv:2509.11395 [math.NT], 2025. See p. 17.

Example

Consider 4 people in a circle in order 1,2,3,4. In the first round, person 1 is skipped, then person 2 is eliminated, then person 3 is skipped. Now people are in order 4,1,3. In the second round, person 4 is skipped, person 1 is eliminated, and person 3 is skipped. Now people are in order 4,3. In the third round person 3 is eliminated. Person 4 is freed. Thus, a(4) = 4.

Prog

(Python)
def J(n, A):
    l=[]
    for i in range(n):
        l.append(i+1)
    index = 0
    P=[]
    for i in range(n):
        index+=A[i]
        index=index%len(l)
        P.append(l[index])
        l.pop(index)
    return P
def invPerm(p):
    inv = []
    for i in range(len(p)):
        inv.append(None)
    for i in range(len(p)):
        inv[p[i]-1]=i+1
    return inv
def survivor(n, A):
    return J(n, A)[n-1]
def UDU(n):
    return [1] + [2 for i in range(n)]
seq = []
for i in range(1, 20):
    seq += [survivor(i, UDU(i))]
print(", ".join([str(v) for v in seq]))
(Python)
def a(n):
    c, i, J = 1, 0, list(range(1, n+1))
    while len(J) > 1:
        i = (i + 1)%len(J)
        q = J.pop(i)
        i = (i + 1)%len(J) # skip the third
    return J[0]
print([a(n) for n in range(1, 73)]) # Michael S. Branicky, Mar 24 2025

Crossrefs

Cf. A006257, A225381, A321298, A378635, A382354, A382356, A382358.

Sequence in context: A317698 A179103 A045814 * A064218 A228780 A333328

Adjacent sequences: A382352 A382353 A382354 * A382356 A382357 A382358

Keyword

nonn

Author

Tanya Khovanova, Nathan Sheffield, and the MIT PRIMES STEP junior group, Mar 22 2025

Status

approved