A380202 Number of card moves to deal n cards using the SpellUnder-Down dealing.

4, 8, 14, 19, 24, 28, 34, 40, 45, 49, 56, 63, 72, 81, 89, 97, 107, 116, 125, 132, 142, 152, 164, 175, 186, 196, 208, 220, 231, 238, 248, 258, 270, 281, 292, 302, 314, 326, 337, 343, 352, 361, 372, 382, 392, 401, 412, 423, 433, 439, 448, 457, 468, 478, 488, 497, 508, 519, 529, 535, 544, 553, 564, 574, 584, 593, 604, 615, 625

Offset

1,1

Comments

In SpellUnder-Down dealing, we spell the number of the next card, putting a card under for each letter in the number, then we deal the next card. So we start with putting 3 cards under, for O-N-E, then deal, then 3 under for T-W-O, then deal, then 5 under for T-H-R-E-E, then deal. The dealing sequence is highly irregular because it depends on English spelling. The dealing pattern starts: UUUDUUUDUUUUUD .

Links

Table of n, a(n) for n=1..69.

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.

Formula

a(n) = A067278(n) + n.

Example

The dealing pattern to deal three cards is UUUDUUUDUUUUUD. It contains 14 letters, thus, a(3) = 14.

Prog

(Python)
from num2words import num2words
from itertools import count, islice
def n2w(n): return "".join(c for c in num2words(n).replace(" and", "") if c.isalpha())
def agen():
    tot = 0
    for n in count(1):
        tot += len(n2w(n))
        yield tot + n
print(list(islice(agen(), 69))) # Michael S. Branicky, Jun 02 2026

Crossrefs

Cf. A005589, A006257, A067278, A225381, A321298, A378635, A380201, A380204, A380246, A380247, A380248.

Sequence in context: A312619 A312620 A312621 * A312622 A312623 A312624

Adjacent sequences: A380199 A380200 A380201 * A380203 A380204 A380205

Keyword

nonn,word,changed

Author

Tanya Khovanova and the MIT PRIMES STEP junior group, Jan 16 2025

Extensions

a(20)-a(69) corrected by Falk Hüffner, May 24 2026

Status

approved