[math-fun] Flipping 4 Penney's game sequences

15 Oct 2010

      This is from Ed Pegg.  --Rich

--------------

I managed to calculate sequences for all of the 4 flip games, in  
Penney's game.

http://en.wikipedia.org/wiki/Penney's_game

The count of ways HHHH beats HHHT on move 4, 5, 6, 7 ... leads to the  
sequence     
1,1,2,4,7,13,24,44,81,149,274,504,927,1705,3136,5768,10609,19513
This is A000073 , the Tribonacci numbers.

In the pasted list below, all the various types of 4-flip games are  
listed along with the odds of winning, a generating function or  
recurrence equation, and 30 terms of the sequence.  If anyone would  
like it as a nicely formatted text file, write to me.

Outlier:  HHTT beats HTHT  seems excessively complicated.

HHHH beats TTTH has a chance of winning of 7/22, or about 1/Pi.  It's A123908.

--Ed Pegg Jr

{
A000012 {HHHT beats THHH,1/8,  (1-n)^(-1),                              

{1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}},
A000071 {HHHT beats TTTH,1/2,  n/(1-2n+n^3),                            

{1,2,4,7,12,20,33,54,88,143,232,376,609,986,1596,2583,4180,6764,10945,17710,28656,46367,75024,121392,196417,317810,514228,832039,1346268,2178308,3524577,5702886,9227464,14930351}},
A000073 {HTTT beats TTTH,7/8,  (1-n(1+n+n^2))^(-1),                     

{1,2,4,7,13,24,44,81,149,274,504,927,1705,3136,5768,10609,19513,35890,66012,121415,223317,410744,755476,1389537,2555757,4700770,8646064,15902591,29249425,53798080,98950096}},
A000073 {HHHH beats HHHT,1/2,  -(n/(-1+n+n^2+n^3)),                     

{1,1,2,4,7,13,24,44,81,149,274,504,927,1705,3136,5768,10609,19513,35890,66012,121415,223317,410744,755476,1389537,2555757,4700770,8646064,15902591,29249425,53798080,98950096}},
A003476 {HTHT beats THHH,7/16, (1+n+2n^2)/(2-2n-4n^3),                  

{1,2,3,5,9,15,25,43,73,123,209,355,601,1019,1729,2931,4969,8427,14289,24227,41081,69659,118113,200275,339593,575819,976369,1655555,2807193,4759931,8071041,13685427,23205289}},
A003479 {HTHT beats HTTT,1/2,  n/((-1+n)(-1+n+2n^3)),                   

{1,2,3,6,11,18,31,54,91,154,263,446,755,1282,2175,3686,6251,10602,17975,30478,51683,87634,148591,251958,427227,724410,1228327,2082782,3531603,5988258,10153823,17217030,29193547}},
A005251 {HHTH beats TTTT,5/8,  (1+(-1+n)n)/(1+n(-2+n-n^2)),             

{1,2,4,7,12,21,37,65,114,200,351,616,1081,1897,3329,5842,10252,17991,31572,55405,97229,170625,299426,525456,922111,1618192,2839729,4983377,8745217,15346786,26931732,47261895}},
A005314 {HHHT beats HTHH,1/2,  n/(1+n(-2+n-n^2)),                       

{1,2,3,5,9,16,28,49,86,151,265,465,816,1432,2513,4410,7739,13581,23833,41824,73396,128801,226030,396655,696081,1221537,2143648,3761840,6601569,11584946,20330163,35676949}},
A078044 {HTHT beats HHHT,3/8,  -((n(1+n))/(-1+n+2n^3)),                 

{1,2,2,4,8,12,20,36,60,100,172,292,492,836,1420,2404,4076,6916,11724,19876,33708,57156,96908,164324,278636,472452,801100,1358372,2303276,3905476,6622220,11228772,19039724}},
A005251 {HHHH beats THHT,1/4,  {-a(n)+a(n-1)+a(n-2)+a(n-4)=0,           
  a(1)=1,a(2)=1,a(3)= 2,a(4)=4},                  
{1,1,2,4,7,12,21,37,65,114,200,351,616,1081,1897,3329,5842,10252,17991,31572,55405,97229,170625,299426,525456,922111,1618192,2839729,4983377,8745217,15346786,26931732}},
A014743 {HHTH beats HHHH,3/5,  {-a(n)-a(1+n)-a(3+n)-a(4+n)+a(5+n)=0,    
  a(1)=1,a(2)=2,a(3)=3,a(4)=5,a(5)=9},            
{1,2,3,5,9,17,31,56,101,183,332,602,1091,1977,3583,6494,11770,21332,38662,70071,126997,230170,417161,756064,1370293,2483524,4501148,8157897,14785402,26797116,48567190}},
A077930 {HHTH beats HHHT,1/3,  {-a(n)-a(2+n)-a(3+n)+a(4+n)=0,           
  a(1)=1,a(2)=1,a(3)=2,a(4)=3},                   
{1,1,2,3,6,10,18,31,55,96,169,296,520,912,1601,2809,4930,8651,15182,26642,46754,82047,143983,252672,443409,778128,1365520,2396320,4205249,7379697,12950466,22726483,39882198}},
A079959 {HHHH beats HTTH,4/11, {-a(n)-a(2+n)-a(4+n)-a(5+n)+a(6+n)=0,    
  a(1)=1,a(2)=1,a(3)=2,a(4)=3,a(5)=6,a(6)=10},    
{1,1,2,3,6,10,19,33,60,106,191,340,610,1089,1950,3485,6236,11150,19946,35670,63802,114107,204091,365018,652857,1167652,2088402,3735179,6680529,11948378,21370166,38221375}},
A079976 {HHHH beats HHTH,2/5,  {-a(n)-a(1+n)-a(3+n)-a(4+n)+a(5+n)=0,    
  a(1)=1,a(2)=1,a(3)=2,a(4)=3,a(5)=6},            
{1,1,2,3,6,11,20,36,65,118,214,388,703,1274,2309,4185,7585,13747,24915,45156,81841,148329,268832,487232,883061,1600463,2900685,5257212,9528190,17268926,31298264,56725087}},
A116732 {HHTT beats HHHT,1/3,  { a(n)-a(1+n)-a(2+n)-a(3+n)+a(4+n)=0,    
  a(1)=1,a(2)=1,a(3)=2,a(4)=4},                   
{1,1,2,4,6,11,19,32,56,96,165,285,490,844,1454,2503,4311,7424,12784,22016,37913,65289,112434,193620,333430,574195,988811,1702816,2932392,5049824,8696221,14975621,25789274}},
A123908 {HHHH beats TTTH,7/22, {-a(n)-2a(1+n)-2a(2+n)-a(3+n)+a(5+n)=0,  
  a(1)=1,a(2)=1,a(3)=2,a(4)=3,a(5)=6},            
{1,1,2,3,6,10,17,30,52,90,156,271,470,815,1414,2453,4255,7381,12804,22211,38529,66836,115940,201120,348881,605201,1049837,1821143,3159121,5480100,9506282,16490465,28605867}},
A128588 {HHTH beats TTTH,7/16, {-a(n)-a(1+n)+a(2+n)=0,                  
  a(1)=1,a(2)=2,a(3)=4},                          
{1,2,4,6,10,16,26,42,68,110,178,288,466,754,1220,1974,3194,5168,8362,13530,21892,35422,57314,92736,150050,242786,392836,635622,1028458,1664080,2692538,4356618,7049156,11405774}},
         {HHTT beats TTTH,7/12, -((n(1+n+n^2))/(-1+n+n^2+n^3-n^4)),     

{1,2,4,7,12,21,36,62,107,184,317,546,940,1619,2788,4801,8268,14238,24519,42224,72713,125218,215636,371343,639484,1101245,1896436,3265822,5624019,9685032,16678437,28721666}},
         {HTHH beats THHH,7/12, -((n(1+n+n^2))/(-1+n+n^2+n^4)),         

{1,2,4,6,11,19,34,59,104,182,320,561,985,1728,3033,5322,9340,16390,28763,50475,88578,155443,272784,478702,840064,1474209,2587057,4539968,7967089,13981266,24535412,43056646}},
         {HHHT beats THTH,9/16, (1+n^2)/(2(-1+n)(-1+n+2n^3)),           

{1,2,4,7,12,21,36,61,104,177,300,509,864,1465,2484,4213,7144,12113,20540,34829,59056,100137,169796,287909,488184,827777,1403596,2379965,4035520,6842713,11602644,19673685,33359112}},
         {HTTT beats HHHT,3/7,  (n(1+n))/((-1+n)(-1+n+n^2)(1+n+n^2)),   

{1,2,3,6,10,16,27,44,71,116,188,304,493,798,1291,2090,3382,5472,8855,14328,23183,37512,60696,98208,158905,257114,416019,673134,1089154,1762288,2851443,4613732,7465175}},
         {HHHT beats HTHT,5/8,  (n+n^3)/((-1+n)(-1+n+2n^3)),            

{1,2,4,8,14,24,42,72,122,208,354,600,1018,1728,2930,4968,8426,14288,24226,41080,69658,118112,200274,339592,575818,976368,1655554,2807192,4759930,8071040,13685426,23205288}},
         {HTTT beats TTHT,7/12, (n-n^3+n^4)/(1-2n+n^3-n^4+n^5),         

{1,2,3,6,11,20,35,62,109,192,337,592,1039,1824,3201,5618,9859,17302,30363,53284,93507,164094,287965,505344,886817,1556256,2731039,4792640,8410497,14759394,25900931,45452966}},
         {HHHT beats HHTT,2/3,  n/(1-2n+2n^4-n^5),                      

{1,2,4,8,14,25,44,76,132,228,393,678,1168,2012,3466,5969,10280,17704,30488,52504,90417,155706,268140,461760,795190,1369385,2358196,4061012,6993404,12043228,20739449,35715070}},
         {HHHT beats HHTH,2/3,  n/(1-2n+n^3-n^4+n^5),                   

{1,2,4,7,13,23,41,72,127,223,392,688,1208,2120,3721,6530,11460,20111,35293,61935,108689,190736,334719,587391,1030800,1808928,3174448,5570768,9776017,17155714,30106180,52832663}},
         {HHTH beats THHH,5/12, -((n(-1+n^2+n^3))/(1-2n+n^3-n^4+n^5)),  

{1,2,3,4,7,12,21,36,63,110,193,338,593,1040,1825,3202,5619,9860,17303,30364,53285,93508,164095,287966,505345,886818,1556257,2731040,4792641,8410498,14759395,25900932,45452967}},
         {HHHH beats TTHH,1/4,  { a(n)-a(2+n)-a(3+n)-a(4+n)+a(5+n)=0,   
   a(1)=1,a(2)=1,a(3)=1,a(4)=2,a(5)=4},            
{1,1,1,2,4,6,11,20,35,62,111,197,350,623,1108,1970,3504,6232,11083,19711,35056,62346,110881,197200,350716,623741,1109311,1972887,3508739,6240221,11098106,19737755,35103195}},
         {HHTT beats HHHH,3/5,  { a(n)-a(2+n)-a(3+n)-a(4+n)+a(5+n)=0,   
   a(1)=1,a(2)=2,a(3)=3,a(4)=6,a(5)=11},           
{1,2,3,6,11,19,34,61,108,192,342,608,1081,1923,3420,6082,10817,19238,34214,60849,108219,192465,342295,608765,1082676,1925517,3424493,6090391,10831636,19263844,34260354,60931341}},
         {HHHH beats HHTT,2/5,  { a(n)-a(2+n)-a(3+n)-a(4+n)+a(5+n)=0,   
   a(1)=1,a(2)=1,a(3)=2,a(4)=4,a(5)=7},            
{1,1,2,4,7,12,22,39,69,123,219,389,692,1231,2189,3893,6924,12314,21900,38949,69270,123195,219100,389665,693011,1232506,2191987,3898404,6933232,12330612,21929742,39001599,69363549}},
         {HHTT beats TTTT,3/4,  { a(n)-a(2+n)-a(3+n)-a(4+n)+a(5+n)=0,   
   a(1)=1,a(2)=2,a(3)=4,a(4)=8,a(5)=14},           
{1,2,4,8,14,25,45,80,142,253,450,800,1423,2531,4501,8005,14237,25320,45031,80087,142433,253314,450514,801230,1424971,2534282,4507169,8015908,14256129,25354235,45091990,80195185}},
         {HTHT beats HHTT,4/9,   
{(-5-n)a(n)+(-3-n)a(2+n)+(2+n)a(3+n)=0,  a(1)=1,a(2)=2,a(3)=3},         

{1,2,3,6,11,18,30,50,81,130,208,330,520,816,1275,1984,3077,4758,7337,11286,17322,26532,40563,61908,94336,143540,218112,331008,501749,759726,1149159,1736534,2621751,3954826,5960902}},
         {HHHT beats TTHH,5/12, {1+a(n)-a(1+n)-a(2+n)-a(3+n)+a(4+n)=0,  
   a(1)=1,a(2)=2,a(3)=3,a(4)=5},                   
{1,2,3,5,8,13,22,37,63,108,185,318,547,941,1620,2789,4802,8269,14239,24520,42225,72714,125219,215637,371344,639485,1101246,1896437,3265823,5624020,9685033,16678438,28721667}},
         {HHHH beats HTTT,4/11,  
{-1-a(n)-2a(1+n)-2a(2+n)-a(3+n)+a(5+n)=0,a(1)=1,a(2)=1,a(3)=2,a(4)=4,a(5)=7},   

{1,1,2,4,7,12,21,37,64,111,193,335,581,1008,1749,3034,5263,9130,15838,27474,47659,82674,143414,248779,431555,748615,1298616,2252698,3907736,6778716,11758980,20398201,35384583}},
         {HHTT beats THHT,5/12, {-1-a(n)-a(2+n)-a(3+n)+a(4+n)=0,        
   a(1)=1,a(2)=1,a(3)=2,a(4)=4},                   
{1,1,2,4,8,14,25,44,78,137,241,423,743,1304,2289,4017,7050,12372,21712,38102,66865,117340,205918,361361,634145,1112847,1952911,3427120,6014177,10554145,18521234,32502500,57037912}},
         {HHHT beats HTTH,4/7,  {-1-a(n)-a(3+n)-a(4+n)+a(5+n)=0,        
   a(1)=1,a(2)=2,a(3)=4,a(4)=7,a(5)=12},           
{1,2,4,7,12,21,36,62,106,181,309,527,899,1533,2614,4457,7599,12956,22089,37660,64207,109467,186631,318188,542480,924876,1576824,2688332,4583345,7814158,13322380,22713363,38724076}},
         {HTTT beats THHT,1/2,  {-1-a(n)-a(3+n)-a(4+n)+a(5+n)=0,        
   a(1)=1,a(2)=2,a(3)=3,a(4)=6,a(5)=10},           
{1,2,3,6,10,18,31,53,91,155,265,452,771,1315,2242,3823,6518,11113,18947,32303,55074,93896,160084,272928,465316,793319,1352532,2305936,3931397,6702650,11427367,19482550,33215854}},
         {HHTT beats HTHT,5/9,   
{-2a(n)+(-4-n)a(1+n)-2a(2+n)-na(3+n)+(1+n)a(4+n)=0,   
a(1)=1,a(2)=2,a(3)=4,a(4)=8},       
{1,2,4,8,14,24,41,68,111,180,289,460,728,1146,1795,2800,4352,6742,10414,16044,24659,37818,57885,88440,134899,205448,312448,474548,719861,1090734,1650908,2496260,3770910,5691360}},
         {HHTH beats HTTH,1/2,  {-2a(n)+2a(1+n)-2a(3+n)+a(4+n)=0,       
   a(1)=1,a(2)=2,a(3)=4,a(4)=6},                   
{1,2,4,6,10,16,28,48,84,144,248,424,728,1248,2144,3680,6320,10848,18624,31968,54880,94208,161728,277632,476608,818176,1404544,2411136,4139136,7105536,12197888,20939776,35946752}},
         {HTHH beats TTHT,7/16, {-2a(n)+2a(1+n)-2a(3+n)+a(4+n)=0,       
   a(1)=1,a(2)=2,a(3)=3,a(4)=5},                   
{1,2,3,5,8,14,24,42,72,124,212,364,624,1072,1840,3160,5424,9312,15984,27440,47104,80864,138816,238304,409088,702272,1205568,2069568,3552768,6098944,10469888,17973376,30854400}},
         {HHTH beats THTT,9/16, {-2a(n)+2a(1+n)-2a(3+n)+a(4+n)=0,       
   a(1)=1,a(2)=2,a(3)=4,a(4)=7},                   
{1,2,4,7,12,20,34,58,100,172,296,508,872,1496,2568,4408,7568,12992,22304,38288,65728,112832,193696,332512,570816,979904,1682176,2887744,4957312,8510080,14609024,25078912,43052288}},
         {HHHT beats THHT,9/16, {-2-a(n)-a(1+n)+a(2+n)=0,               
   a(1)=1,a(2)=2,a(3)=4},                          
{1,2,4,8,14,24,40,66,108,176,286,464,752,1218,1972,3192,5166,8360,13528,21890,35420,57312,92734,150048,242784,392834,635620,1028456,1664078,2692536,4356616,7049154,11405772,18454928}},
         {HTTT beats HHHH,7/11,  
{-3-a(n)-2a(1+n)-2a(2+n)-a(3+n)+a(5+n)=0,a(1)=1,a(2)=2,a(3)=4,a(4)=7,a(5)=13},  

{1,2,4,7,13,23,40,70,122,212,368,639,1109,1924,3338,5791,10046,17427,30231,52442,90971,157807,273747,474867,823748,1428949,2478786,4299929,7459050,12939150,22445432,38935897,67541764}},
         {HHHT beats  
TTTT,15/22,{-4-a(n)-2a(1+n)-2a(2+n)-a(3+n)+a(5+n)=0,a(1)=1,a(2)=2,a(3)=4,a(4)=8,a(5)=14},           
{1,2,4,8,14,25,44,77,134,233,405,703,1220,2117,3673,6372,11054,19176,33265,57705,100101,173645,301221,522526,906422,1572363,2727565,4731484,8207665,14237766,24698130,42843633}},
         {HHHT beats HTTT,4/7,  {-4-n-a(n)-2a(1+n)-a(2+n)+a(4+n)=0,     
   a(1)=1,a(2)=2,a(3)=4,a(4)=8},                   
{1,2,4,8,14,24,41,68,112,184,300,488,793,1286,2084,3376,5466,8848,14321,23176,37504,60688,98200,158896,257105,416010,673124,1089144,1762278,2851432,4613721,7465164,12078896,19544072}},
         {HHTH beats THTH,9/16, {-a(n)+a(1+n)-2a(2+n)+a(3+n)=0,         
   a(1)=1,a(2)=2,a(3)=4,a(4)=6},                   
{1,2,4,6,10,18,32,56,98,172,302,530,930,1632,2864,5026,8820,15478,27162,47666,83648,146792,257602,452060,793310,1392162,2443074,4287296,7523680,13203138,23169892,40660326,71353898}},
         {HTHT beats THHT,7/16, {-a(n)+a(1+n)-2a(2+n)+a(3+n)=0,         
   a(1)=1,a(2)=2,a(3)=2,a(4)=4},                   
{1,2,2,4,8,14,24,42,74,130,228,400,702,1232,2162,3794,6658,11684,20504,35982,63144,110810,194458,341250,598852,1050912,1844222,3236384,5679458,9966754,17490434,30693572,53863464}},
         {HHTH beats THHT,5/12, {-a(n)+a(2+n)-2a(4+n)+a(5+n)=0,         
   a(1)=1,a(2)=1,a(3)=2,a(4)=4,a(5)=7},            
{1,1,2,4,7,13,23,41,73,130,232,414,739,1319,2354,4201,7497,13379,23876,42609,76040,135701,242172,432180,771268,1376404,2456329,4383562,7822900,13960739,24914320,44462069,79346961}},
         {HTHH beats THHT,7/12, {-a(n)+a(2+n)-2a(4+n)+a(5+n)=0,         
   a(1)=1,a(2)=2,a(3)=3,a(4)=6,a(5)=10},           
{1,2,3,6,10,18,32,57,102,182,325,580,1035,1847,3296,5882,10497,18733,33431,59661,106471,190008,339088,605136,1079925,1927233,3439338,6137839,10953581,19547749,34884892,62255541}},
         {HHTT beats HTHH,3/7,  {-a(n)+a(3+n)-2a(5+n)+a(6+n)=0,         
   a(1)=1,a(2)=2,a(3)=3,a(4)=5,a(5)=8,a(6)=13},    
{1,2,3,5,8,13,22,38,66,115,200,347,601,1040,1799,3112,5384,9316,16121,27898,48279,83549,144584,250205,432982,749278,1296630,2243827,3882960,6719495,11628145,20122608,34822351}},
         {HHTT beats TTHT,9/14, {-a(n)+a(3+n)-2a(5+n)+a(6+n)=0,         
   a(1)=1,a(2)=2,a(3)=4,a(4)=8,a(5)=14,a(6)=24},   
{1,2,4,8,14,24,41,70,120,207,358,620,1074,1860,3220,5573,9644,16688,28877,49970,86472,149640,258954,448124,775485,1341986,2322320,4018795,6954558,12034920,20826530,36040488}},
         {HHTH beats TTHH,5/14, {-a(n)+a(3+n)-2a(5+n)+a(6+n)=0,         
   a(1)=1,a(2)=2,a(3)=3,a(4)=4,a(5)=6,a(6)=9},     
{1,2,3,4,6,9,15,26,46,81,142,247,428,740,1279,2211,3824,6616,11449,19814,34291,59344,102698,177721,307547,532210,920990,1593777,2758042,4772815,8259400,14292968,24734111,42802599}},
         {HTHH beats HHTT,4/7,  {-a(n)+a(3+n)-2a(5+n)+a(6+n)=0,         
   a(1)=1,a(2)=2,a(3)=4,a(4)=7,a(5)=12,a(6)=20},   
{1,2,4,7,12,20,34,58,100,173,300,520,901,1560,2700,4672,8084,13988,24205,41886,72484,125435,217068,375640,650050,1124918,1946680,3368745,5829640,10088240,17457785,30210848}},
         {HHHH beats HTHT,5/12, {-a(n)-2a(1+n)-a(3+n)+a(4+n)=0,         
   a(1)=1,a(2)=1,a(3)=2,a(4)=4},                   
{1,1,2,4,7,12,22,40,71,127,229,411,736,1321,2372,4255,7633,13698,24580,44101,79130,141988,254770,457131,820237,1471765,2640797,4738402,8502169,15255528,27373129,49115869,88129094}},
         {HTHT beats HHHH,7/12, {-a(n)-2a(1+n)-a(3+n)+a(4+n)=0,         
   a(1)=1,a(2)=2,a(3)=3,a(4)=5},                   
{1,2,3,5,10,18,31,56,102,182,325,585,1051,1883,3378,6065,10882,19521,35029,62858,112782,202361,363106,651528,1169032,2097605,3763767,6753359,12117601,21742740,39013225,70001786}},
         {HTHT beats TTTT,5/8,  {-a(n)-2a(1+n)-a(3+n)+a(4+n)=0,         
   a(1)=1,a(2)=2,a(3)=3,a(4)=6},                   
{1,2,3,6,11,19,34,62,111,198,356,640,1147,2057,3693,6627,11888,21331,38278,68681,123231,221118,396758,711901,1277368,2292002,4112562,7379199,13240571,23757697,42628657,76488998}},
         {HHHH beats THTH,3/8,  {-a(n)-2a(1+n)-a(3+n)+a(4+n)=0,         
   a(1)=1,a(2)=1,a(3)=2,a(4)=3},                   
{1,1,2,3,6,11,19,34,62,111,198,356,640,1147,2057,3693,6627,11888,21331,38278,68681,123231,221118,396758,711901,1277368,2292002,4112562,7379199,13240571,23757697,42628657,76488998}},
         {HTTT beats TTTT,15/16,{-a(n)-a(1+n)-a(2+n)+a(3+n)=0,          
   a(1)=1,a(2)=2,a(3)=4,a(4)=8},                   
{1,2,4,8,14,26,48,88,162,298,548,1008,1854,3410,6272,11536,21218,39026,71780,132024,242830,446634,821488,1510952,2779074,5111514,9401540,17292128,31805182,58498850,107596160}},
         {HHHH beats HTHH,3/10, {-a(n)-a(1+n)-a(3+n)-a(4+n)+a(5+n)=0,   
   a(1)=1,a(2)=1,a(3)=1,a(4)=2,a(5)=4},            
{1,1,1,2,4,8,14,25,45,82,149,270,489,886,1606,2911,5276,9562,17330,31409,56926,103173,186991,338903,614229,1113231,2017624,3656749,6627505,12011714,21770074,39456161,71510489}},
         {HTHH beats HHHH,7/10, {-a(n)-a(1+n)-a(3+n)-a(4+n)+a(5+n)=0,   
   a(1)=1,a(2)=2,a(3)=4,a(4)=6,a(5)=11},           
{1,2,4,6,11,20,37,67,121,219,397,720,1305,2365,4286,7768,14079,25517,46247,83818,151912,275326,499002,904393,1639125,2970756,5384209,9758360,17686087,32054328,58095380,105292277}},
         {HTTH beats HHHH,7/11, {-a(n)-a(2+n)-a(4+n)-a(5+n)+a(6+n)=0,   
   a(1)=1,a(2)=2,a(3)=4,a(4)=6,a(5)=11,a(6)=19},   
{1,2,4,6,11,19,35,62,112,199,357,637,1141,2039,3649,6524,11671,20871,37332,66766,119418,213579,382000,683216,1221966,2185527,3908911,6991233,12504110,22364086,39999073,71539919}},
         {HHTH beats HTHT,5/7,  {-a(n)-a(3+n)+a(4+n)-2a(5+n)+a(6+n)=0,  
   a(1)=1,a(2)=2,a(3)=4,a(4)=7,a(5)=12,a(6)=21},   
{1,2,4,7,12,21,38,69,125,226,408,736,1328,2397,4327,7811,14100,25452,45943,82931,149698,270219,487771,880473,1589337,2868903,5178640,9347933,16873900,30458980,54981330,99246483}},
         {HTHT beats HHTH,2/7,  {-a(n)-a(3+n)+a(4+n)-2a(5+n)+a(6+n)=0,  
   a(1)=1,a(2)=1,a(3)=1,a(4)=2,a(5)=4,a(6)=7},     
{1,1,1,2,4,7,13,24,43,77,139,251,453,818,1477,2666,4812,8686,15679,28302,51088,92219,166464,300483,542400,979083,1767337,3190210,5758630,10394870,18763720,33870283,61139053}},
         {HTTH beats HHHT,3/7,  {-a(n)-a(3+n)-a(4+n)+a(5+n)=0,          
   a(1)=1,a(2)=2,a(3)=3,a(4)=5,a(5)=8},            
{1,2,3,5,8,14,24,41,70,119,203,346,590,1006,1715,2924,4985,8499,14490,24704,42118,71807,122424,208721,355849,606688,1034344,1763456,3006521,5125826,8739035,14899205,25401696}},
         {HTHH beats THTH,5/14, {-a(n)-a(3+n)+a(4+n)-2a(5+n)+a(6+n)=0,  
   a(1)=1,a(2)=1,a(3)=2,a(4)=3,a(5)=5,a(6)=9},     
{1,1,2,3,5,9,17,31,56,101,182,328,592,1069,1930,3484,6289,11352,20491,36988,66767,120521,217552,392702,708864,1279566,2309737,4169293,7525967,13585080,24522350,44265153,79902773}},
         {HTTH beats THHH,1/2,  {-a(n)-a(3+n)-a(4+n)+a(5+n)=0,          
   a(1)=1,a(2)=2,a(3)=4,a(4)=6,a(5)=10},           
{1,2,4,6,10,17,29,50,85,145,247,421,718,1224,2087,3558,6066,10342,17632,30061,51251,87378,148971,253981,433013,738245,1258636,2145852,3658469,6237334,10634048,18130018,30909918}},
         {HTHT beats THTT,9/14, {-a(n)-a(3+n)+a(4+n)-2a(5+n)+a(6+n)=0,  
   a(1)=1,a(2)=2,a(3)=3,a(4)=6,a(5)=11,a(6)=19},   
{1,2,3,6,11,19,34,62,112,202,365,659,1189,2146,3874,6993,12623,22786,41131,74245,134019,241917,436683,788254,1422873,2568420,4636240,8368850,15106563,27268770,49222700,88851613}}}

[math-fun] Flipping 4 Penney's game sequences

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