1,3

This variation of the Josephus problem is related to down-under-under card dealing. The n-th row has n elements.

In this variation of the Josephus elimination process, the numbers 1 through n are arranged in a circle. A pointer starts at position 1. With each turn, the pointer eliminates the number and then skips two numbers. The process repeats until no numbers remain. This sequence represents the triangle T(n,k), where n is the number of people in the circle, and T(n,k) is the elimination order of the k-th number in the circle.

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.

Consider 4 people in a circle. Initially, person number 1 is eliminated, and persons 2 and 3 are skipped. The remaining people are now in order 4, 2, 3. Then, person 4 is eliminated, and 2 and 3 are skipped. The remaining people are in order 2, 3. Now, person 2 is eliminated. Person 3 is eliminated last. Thus, the fourth row of the triangle is 1, 4, 2, 3.

Triangle begins:

1;

1, 2;

1, 2, 3,;

1, 4, 2, 3;

1, 4, 3, 5, 2;

1, 4, 2, 6, 3, 5;

1, 4, 7, 5, 3, 6, 2;

1, 4, 7, 3, 8, 6, 2, 5;

...

(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 DUU(n):

return [0] + [2 for i in range(n)]

seq = []

for i in range(1, 20):

seq += J(i, DUU(i))

print(", ".join([str(v) for v in seq]))

(Python)

def row(n):

i, J, out = 0, list(range(1, n+1)), []

while len(J) > 1:

i = i%len(J)

out.append(J.pop(i))

i = (i + 2)%len(J)

return out + [J[0]]

print([e for n in range(1, 14) for e in row(n)]) # Michael S. Branicky, Mar 27 2025

Cf. A006257, A016777, A054995, A225381, A321298, A378635, A381622.

Sequence in context: A143604 A021475 A132224 * A194961 A195110 A167198

Adjacent sequences: A381620 A381621 A381622 * A381624 A381625 A381626

nonn,tabl

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

approved