# [seqfan] Penney's game sequences

Ed Pegg Jr ed at mathpuzzle.com
Fri Oct 15 22:21:32 CEST 2010

```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

```