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

1,2

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

References

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

Links

Table of n, a(n) for n=1..55.

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).

Example

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;
  ...

Prog

(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

Crossrefs

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

Sequence in context: A103884 A103883 A317640 * A183892 A019432 A211796

Adjacent sequences: A125232 A125233 A125234 * A125236 A125237 A125238

Keyword

nonn,tabl,easy

Author

Gary W. Adamson, Nov 24 2006

Extensions

More terms from Michel Marcus, Mar 04 2014

Status

approved