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A381128 The number of card moves required to deal n cards using Down-SpellUnder dealing.

%I A381128 #9 Mar 14 2025 21:13:25
%S A381128 1,5,9,15,20,25,29,35,41,46,50,57,64,73,82,90,98,108,117,126,133,143,
%T A381128 153,165,176,187,197,209,221,232,239,249,259,271,282,293,303,315,327,
%U A381128 338,344,353,362,373,383,393,402,413,424,434,440,449,458,469,479,489,498,509,520,530,536,545,554,565,575,585,594,605
%N A381128 The number of card moves required to deal n cards using Down-SpellUnder dealing.
%C A381128 In Down-SpellUnder dealing, after dealing the i-th card we move a card from the top of the deck to the bottom for each letter in the English spelling of i. Then we deal the next card and proceed likewise. So we start by dealing card 1, then putting 3 cards under because O-N-E has three letters. Then, we deal the next card, and put three cards under 3 because T-W-O has three letters. We then deal again, put 5 under for T-H-R-E-E, and so on. This dealing sequence is highly irregular because it depends on English spelling. The dealing pattern starts: DUUUDUUUDUUUUUD, where D means 'deal', and U means 'under'.
%F A381128 a(n) = A067278(n-1) + n.
%e A381128 The dealing pattern to deal four cards is DUUUDUUUDUUUUUD. It contains 15 letters, so a(4) = 15.
%o A381128 (Python)
%o A381128 from num2words import num2words as n2w
%o A381128 def spell(n):
%o A381128     return sum(1 for c in n2w(n).replace(" and", "").replace(" ", "").replace(",", "").replace("-", ""))
%o A381128 moves_so_far = 0
%o A381128 l = []
%o A381128 for i in range(1, 41):
%o A381128     moves_so_far += 1
%o A381128     l += [str(moves_so_far)]
%o A381128     moves_so_far += spell(i)
%o A381128 print(", ".join(l))
%Y A381128 Cf. A005589, A006257, A225381, A321298, A378635, A380201, A380202, A380204, A380246, A380247, A380248, A381114, A381127, A381129.
%K A381128 nonn,word
%O A381128 1,2
%A A381128 _Tanya Khovanova_, _Nathan Sheffield_, and the MIT PRIMES STEP junior group, Feb 14 2025