A127172 Cube of A051731.

1, 3, 1, 3, 0, 1, 6, 3, 0, 1, 3, 0, 0, 0, 1, 9, 3, 3, 0, 0, 1, 3, 0, 0, 0, 0, 0, 1, 10, 6, 0, 3, 0, 0, 0, 1, 6, 0, 3, 0, 0, 0, 0, 0, 1, 9, 3, 0, 0, 0, 3, 0, 0, 0, 0, 1

Offset: 1

Author

Gary W. Adamson, Jan 06 2007

Keywords

• nonn
• tabl
• uned

Comments

Nonzero terms in every column = A007425: (1, 3, 3, 6, 3, 9, 3, ...).

Row sums = A007426: (1, 4, 4, 20, 4, 16, ...).

A127172 * mu(n) = d(n); or A127172 * A008683 = A000005.

A127172 * d(n) = tau_5(n); or A127172 * A000005 = A061200.

A127172 * phi(n) = A007429: (1, 4, 5, 11, 7, 20, ...); or: A127172 * A000010 = A007429.

Note that A051731 * d(n) = row sums of A127172; or A051731 * A000005 = A007425.

Also, A126988 * mu(n) = phi(n); or A126988 * A008683 = A000010.

A126988 * phi(n) = A018804: (1, 3, 5, 8, 9, 15, ...); = A127170 * mu(n).

Examples

			First few rows of the triangle:
   1;
   3, 1;
   3, 0, 1;
   6, 3, 0, 1;
   3, 0, 0, 0, 1;
   9, 3, 3, 0, 0, 1;
   3, 0, 0, 0, 0, 0, 1;
  10, 6, 0, 3, 0, 0, 0, 1;
   6, 0, 3, 0, 0, 0, 0, 0, 1;
   9, 3, 0, 0, 3, 0, 0, 0, 0, 1;
  ...
		

Crossrefs

Cf. A000005, A007425, A127170, A051731, A007429, A000010, A126988, A018804.

Formula

Cube of A051731 A007425: (1, 3, 3, 6, 3, 9, 3, ...) in every column k, interspersed with (k-1) zeros.