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.

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

Author

Eric W. Weisstein

Keywords

• nonn
• nice
• tabl

Examples

			Triangle begins:
   1;
   3,  1;
   7,  3, 1;
  15,  8, 3, 1;
  31, 19, 8, 3, 1
  ...
		

References

• W. Feller, An Introduction to Probability Theory and Its Applications, Vol. 1, 2nd ed. New York: Wiley, p. 300, 1968.

Links

• T. D. Noe, Rows n=1..100 of triangle, flattened
• Eric Weisstein's World of Mathematics, Coin Tossing.
• Eric Weisstein's World of Mathematics, Run.

Crossrefs

Cf. A008466, A119706 (row sums?).

Cf. A050231, A050232, A050233.

Programs

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 *)