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A050227
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.
3
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, 255, 201, 107, 48, 20, 8, 3, 1, 511, 423, 238, 111, 48, 20, 8, 3, 1, 1023, 880, 520, 251, 112, 48, 20, 8, 3, 1, 2047, 1815, 1121, 558, 255, 112, 48, 20, 8, 3, 1, 4095, 3719
OFFSET
1,2
REFERENCES
W. Feller, An Introduction to Probability Theory and Its Applications, Vol. 1, 2nd ed. New York: Wiley, p. 300, 1968.
LINKS
Eric Weisstein's World of Mathematics, Coin Tossing.
Eric Weisstein's World of Mathematics, Run.
EXAMPLE
Triangle begins:
1;
3, 1;
7, 3, 1;
15, 8, 3, 1;
31, 19, 8, 3, 1
...
MATHEMATICA
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 *)
CROSSREFS
KEYWORD
nonn,nice,tabl
STATUS
approved