A380247 Triangle read by rows: T(n,k) is the number of the k-th eliminated person in the variation of the Josephus elimination process for n people, where the number of people skipped correspond to the number of letters in the next number in English alphabet.

1, 2, 1, 1, 3, 2, 4, 1, 3, 2, 4, 3, 2, 5, 1, 4, 2, 5, 1, 3, 6, 4, 1, 2, 3, 6, 5, 7, 4, 8, 7, 6, 1, 2, 5, 3, 4, 8, 6, 3, 2, 1, 9, 7, 5, 4, 8, 5, 1, 9, 6, 10, 7, 2, 3, 4, 8, 3, 10, 6, 1, 2, 7, 11, 9, 5, 4, 8, 2, 9, 3, 10, 7, 11, 12, 1, 5, 6, 4, 8, 1, 7, 13, 6, 3, 2, 12, 11, 5, 9, 10, 4, 8, 14, 6, 12, 3, 13, 10, 7, 2, 11, 1, 5, 9, 4

Offset

1,2

Comments

In this variation of the Josephus elimination process, the numbers 1 through n are arranged in a circle. A pointer starts at position 1. Then three people are skipped because number O-N-E has three letters, then the next person is eliminated. Next, three people are skipped because T-W-O has three letters, and the next person is eliminated. Then, five people are skipped because T-H-R-E-E has five letters, and so on. This 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 person in the circle.

In rows 4 and after, the first number is 4. In rows 8 and after, the second number is 8. In rows 14 and after, the third number is 14. In the limit the numbers form sequence A380202.

Links

Table of n, a(n) for n=1..106.

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

Triangle begins:
  1;
  2, 1;
  1, 3, 2;
  4, 1, 3, 2;
  4, 3, 2, 5, 1;
  4, 2, 5, 1, 3, 6;
  4, 1, 2, 3, 6, 5, 7;
  ...
For n = 4 suppose four people are arranged in a circle corresponding to the fourth row of the triangle. Three people are skipped for each letter in O-N-E; then the 4th person is eliminated. This means the row starts with 4. The next three people are skipped, and the person eliminated is number 1. Thus, the next element in the row is 1. Then, 5 people are skipped, and the next person eliminated is number 3. Similarly, the last person eliminated is number 2. Thus, the fourth row of this triangle is 4, 1, 3, 2.

Prog

(Python)
from num2words import num2words as n2w
def spell(n):
    return sum(1 for c in n2w(n).replace(" and", "").replace(" ", "").replace(chr(44), "").replace("-", ""))
def inverse_permutation(p):
    inv = [0] * len(p)
    for i, x in enumerate(p):
        inv[x-1] = i +1
    return inv
def nthRow(n):
    l = []
    for i in range(0, n):
        l.append(0)
    zp = 0
    for j in range(1, n+1):
        zc = 0
        while zc <= spell(j):
            if l[zp] == 0:
                zc += 1
            zp += 1
            zp = zp % n
        l[zp-1] = j
    return l
l = []
for i in range(1, 15):
    l += inverse_permutation(nthRow(i))
print(l)
(Python)
from num2words import num2words as n2w
def f(n): return sum(1 for c in n2w(n).replace(" and", "") if c.isalpha())
def row(n):
    c, i, J = 1, 0, list(range(1, n+1))
    out = []
    while len(J) > 1:
        i = (i + f(c))%len(J)
        q = J.pop(i)
        out.append(q)
        c = c+1
    out.append(J[0])
    return out
print([e for n in range(1, 15) for e in row(n)]) # Michael S. Branicky, Feb 15 2025

Crossrefs

Cf. A005589, A006257, A225381, A321298, A378635, A380201, A380202, A380204, A380246, A380248.

Sequence in context: A240853 A363094 A319849 * A384772 A117334 A241472

Adjacent sequences: A380244 A380245 A380246 * A380248 A380249 A380250

Keyword

nonn,word,tabl

Author

Tanya Khovanova and the MIT PRIMES STEP junior group, Jan 17 2025

Status

approved