In Penney's game, http://en.wikipedia.org/wiki/Penney's_game , I have HHT, and you have HHH. We start flipping a coin, and the winner is the first to get their sequence of heads and tails. In HHH beats HHT, the number of wins on turn 3, 4, 5, 6 ... form the Fibonacci sequence. All of the below famous sequences are linked by Penney's game, which doesn't seem to be well known. { {1,1,2,3,5,8,13,21,34,55,89,144,233,377,610,987,1597} (*HHH beats HHT A000045 - Fibonacci*), {1,1,1,2,4,6,9,15,25,40,64,104,169,273,441,714,1156} (*HHH beats HTH A006498*), {1,1,2,3,4,6,8,11,15,20,27,36,48,64,85,113,150} (*HHH beats HTT A023434 - Dying rabbits*), {1,1,2,3,4,6,9,13,19,28,41,60,88,129,189,277,406} (*HHH beats THT A000930*), {1,1,1,2,2,3,4,5,7,9,12,16,21,28,37,49,65} (*HHH beats TTH A000931 - Padovan*), {1,2,3,5,8,12,18,27,40,59,87,128,188,276,405,594,871} (*HHT beats HTH A077868*), {1,2,4,6,9,12,16,20,25,30,36,42,49,56,64,72,81} (*HHT beats HTT A002620 - Quarter squares*), {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1} (*HHT beats THH A000012 - All 1's*), {1,2,4,6,8,10,12,14,16,18,20,22,24,26,28,30,32} (*HHT beats THT A004277*), {1,2,4,6,9,13,18,25,34,46,62,83,111,148,197,262,348} (*HHT beats TTT Not in OEIS*), {1,2,2,3,6,10,15,24,40,65,104,168,273,442,714,1155,1870} (*HTH beats HHH A070550*), {1,1,1,2,3,4,6,9,13,19,28,41,60,88,129,189,277} (*HTH beats HHT A000930*), {1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17} (*HTH beats HTT A000027 - Natural numbers*), {1,2,2,3,5,7,10,15,22,32,47,69,101,148,217,318,466} (*HTH beats THH A097333*), {1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2} (*HTH beats TTH A040000*), {1,2,3,4,6,9,13,19,28,41,60,88,129,189,277,406,595} (*HTH beats TTT A068921 - Tatami mats*), {1,2,3,5,7,10,14,19,26,35,47,63,84,112,149,198,263} (*HTT beats HHH A054405*), {1,1,2,2,3,3,4,4,5,5,6,6,7,7,8,8,9} (*HTT beats HHT A008619*), {1,1,2,4,6,9,14,21,31,46,68,100,147,216,317,465,682} (*HTT beats THT A038718*), {1,2,3,5,8,13,21,34,55,89,144,233,377,610,987,1597,2584} (*HTT beats TTH A000045*), {1,2,4,6,10,16,26,42,68,110,178,288,466,754,1220,1974,3194} (*HTT beats TTT A128588*)} Ed Pegg Jr mathpuzzle.com