OFFSET
0,6
COMMENTS
In the Penney game THTH beats HTHH 9 times out of 14 yet the expected wait time for THTH is 20 while that for HTHH is only 18.
LINKS
Penney Ante, Counterintuitive Probabilities in Coin Tossing, Bay Area Circle for Teachers Summer Workshop. [broken link]
Raymond S. Nickerson, Penney Ante: Counterintuitive Probabilities in Coin Tossing, 2008.
Eric Weisstein's World of Mathematics, Coin Tossing
Wikipedia, Penney's game
Index entries for linear recurrences with constant coefficients, signature (2,-1,1,0,0,1).
FORMULA
G.f.: G(x) = (x^4 + x^7)/(1 - 2x + x^2 - x^3 - x^6). We note G(1/2) = 9/14.
EXAMPLE
a(7)=6 because we have: TTTTHTH, THTTHTH, THHTHTH, HTTTHTH, HHTTHTH, HHHTHTH.
MATHEMATICA
nn=40; CoefficientList[Series[(x^4+x^7)/(1-2x+x^2-x^3-x^6), {x, 0, nn}], x]
LinearRecurrence[{2, -1, 1, 0, 0, 1}, {0, 0, 0, 0, 1, 2, 3, 6}, 50]
CROSSREFS
KEYWORD
nonn
AUTHOR
Geoffrey Critzer, Mar 01 2014
STATUS
approved