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%I #20 Jan 03 2021 00:55:16
%S 1,3,1,7,3,1,15,8,3,1,31,19,8,3,1,63,43,20,8,3,1,127,94,47,20,8,3,1,
%T 255,201,107,48,20,8,3,1,511,423,238,111,48,20,8,3,1,1023,880,520,251,
%U 112,48,20,8,3,1,2047,1815,1121,558,255,112,48,20,8,3,1,4095,3719
%N Triangle of number of n-tosses having a run of r or more heads for a fair coin with r=1 to n across and n=1, 2, ... down.
%D W. Feller, An Introduction to Probability Theory and Its Applications, Vol. 1, 2nd ed. New York: Wiley, p. 300, 1968.
%H T. D. Noe, <a href="/A050227/b050227.txt">Rows n=1..100 of triangle, flattened</a>
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/CoinTossing.html">Coin Tossing.</a>
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/Run.html">Run.</a>
%e Triangle begins:
%e 1;
%e 3, 1;
%e 7, 3, 1;
%e 15, 8, 3, 1;
%e 31, 19, 8, 3, 1
%e ...
%t Clear[fib]; fib[n_, n_] = 1; fib[n_, k_] /; k > n = 0; fib[n_, k_] := fib[n, k] = If[k == 1, 1, Sum[fib[m, k], {m, n - k , n - 1}]]; Table[ 2^n - fib[n + k + 1 , k], {n, 1, 12}, {k, 1, n}] // Flatten (* _Jean-François Alcover_, Jan 15 2013 *)
%Y Cf. A008466.
%Y Cf. A050231, A050232, A050233.
%K nonn,nice,tabl
%O 1,2
%A _Eric W. Weisstein_