login
A125235
Triangle with the partial column sums of the octagonal numbers.
1
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.
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
KEYWORD
nonn,tabl,easy
AUTHOR
Gary W. Adamson, Nov 24 2006
EXTENSIONS
More terms from Michel Marcus, Mar 04 2014
STATUS
approved