A344903 a(n) is the number of optimal strategies for Player I in the Penney-Ante game with strings of length n.

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

Michael De Vlieger, Table of n, a(n) for n = 3..3326

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

Adjacent sequences: A344900 A344901 A344902 * A344904 A344905 A344906

Keyword

nonn,easy

Author

Michael De Vlieger, Jun 02 2021

Status

approved