A396158 Triangle; T(n,m) is the number of possible outcomes having a run of m or more sixes when a die is rolled n times.

1, 6, 1, 36, 11, 1, 216, 91, 11, 1, 1296, 671, 96, 11, 1, 7776, 4651, 751, 96, 11, 1, 46656, 31031, 5531, 756, 96, 11, 1, 279936, 201811, 39186, 5611, 756, 96, 11, 1, 1679616, 1288991, 270241, 40091, 5616, 756, 96, 11, 1, 10077696, 8124571, 1827071, 278946, 40171, 5616, 756, 96, 11, 1

Offset

0,2

Comments

See also A109435, the number of possible outcomes having a run of m or more heads when a coin is tossed n times.

Links

Table of n, a(n) for n=0..54.

Formula

T(n,m) = A081041(n-m) for ceiling(n/2) <= m <= n.

T(2*n+1,n) = A081041(n+1) - 5.

T(n,0) = A000400(n).

T(n,1) = A005062(n) = 6^n - A000351(n).

T(n,2) = A395827(n) = 6^n - A180033(n).

T(n,3) = A396152(n).

T(n,4) = A396153(n).

T(n,5) = A396154(n).

T(n,6) = A396155(n).

T(n,7) = A396156(n).

Example

Triangle begins:
  n\m      0       1       2       3       4       5       6       7       8       9
  0        1
  1        6       1
  2       36      11       1
  3      216      91      11       1
  4     1296     671      96      11       1
  5     7776    4651     751      96      11       1
  6    46656   31031    5531     756      96      11       1
  7   279936  201811   39186    5611     756      96      11       1
  8  1679616 1288991  270241   40091    5616     756      96      11       1
  9 10077696 8124571 1827071  278946   40171    5616     756      96      11       1

Crossrefs

Cf. A000351, A000400, A005062, A081041, A180033, A395827, A396152, A396153, A396154, A396155, A396156, A109435.

Sequence in context: A051930 A347488 A147320 * A038255 A075501 A089504

Adjacent sequences: A396155 A396156 A396157 * A396159 A396160 A396161

Keyword

nonn,tabl

Author

A.H.M. Smeets, May 21 2026

Extensions

a(47) corrected by Sean A. Irvine, Jun 07 2026

Status

approved