A125235 Triangle with the partial column sums of the octagonal numbers.

1, 8, 1, 21, 9, 1, 40, 30, 10, 1, 65, 70, 40, 11, 1, 96, 135, 110, 51, 12, 1, 133, 231, 245, 161, 63, 13, 1, 176, 364, 476, 406, 224, 76, 14, 1, 225, 540, 840, 882, 630, 300, 90, 15, 1, 280, 765, 1380, 1722, 1512, 930, 390, 105, 16, 1

Offset: 1

Author

Gary W. Adamson, Nov 24 2006

Keywords

• nonn
• tabl
• easy

Comments

"Partial column sums" means the octagonal numbers are the 1st column, the 2nd column are the partial sums of the 1st column, the 3rd column are the partial sums of the 2nd, etc.

Row sums are 1, 9, 31, 81, 187, 405, 847 = 7*(2^n-1) - 6*n. - R. J. Mathar, Sep 06 2011

Examples

			First few rows of the triangle:
   1;
   8,   1;
  21,   9,   1;
  40,  30,  10,   1;
  65,  70,  40,  11,   1;
  96, 135, 110,  51,  12,   1;
  ...
		

References

• Albert H. Beiler, Recreations in the Theory of Numbers, Dover (1966), p. 189.

Crossrefs

Cf. A000567, A002414, A002419, A051843, A027810, A125232 - A125234.

Programs

PARI

t(n, k) = if (n <0, 0, if (k==1, n*(3*n-2), if (k > 1, t(n-1,k-1) + t(n-1,k))));
tabl(nn) = {for (n = 1, nn, for (k = 1, n, print1(t(n, k), ", ");); print(););} \\ Michel Marcus, Mar 04 2014

Formula

T(n,1) = A000567(n).

T(n,k) = T(n-1,k-1) + T(n-1,k), k>1.

T(n,2) = A002414(n-1).

T(n,3) = A002419(n-2).

T(n,4) = A051843(n-4).

T(n,5) = A027810(n-6).

Extensions

More terms from Michel Marcus, Mar 04 2014