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A344903
a(n) is the number of optimal strategies for Player I in the Penney-Ante game with strings of length n.
1
4, 2, 2, 2, 6, 10, 22, 42, 86, 166, 338, 666, 1342, 2662, 5346, 10650, 21342, 42598, 85282, 170398, 340962, 681586, 1363510, 2726354, 5453374, 10905406, 21812154, 43621646, 87245954, 174486562, 348978470, 697946290, 1395903230, 2791785118, 5583591578, 11167140558
OFFSET
3,1
COMMENTS
See p. 5 of Phillips paper for details.
LINKS
Reed Phillips and A. J. Hildebrand, The number of optimal strategies in the Penney-Ante game, Integers (2021) Vol. 21, #A27.
Wikipedia, Penney's game
FORMULA
a(3) = 4, a(4) = a(5) = 2, a(n) = 2 * a(n-1) - (-1)^n * a(Floor(n/2)+1) for n >= 6.
MATHEMATICA
Block[{a}, a[3] = 4; a[4] = a[5] = 2; a[n_] := 2 a[n - 1] - (-1)^n*a[Floor[n/2] + 1]; Array[a[#] &, 36, 3]]
(* Second program, faster: *)
Block[{a = {0, 0, 4, 2, 2}}, Do[AppendTo[a, 2 a[[i - 1]] - (-1)^i*a[[Floor[i/2] + 1]]], {i, 6, 38}]; Drop[a, 2]] (* Michael De Vlieger, Jun 04 2021 *)
CROSSREFS
Sequence in context: A341686 A353636 A255909 * A198101 A364686 A341859
KEYWORD
nonn,easy
AUTHOR
Michael De Vlieger, Jun 02 2021
STATUS
approved