

A236112


Triangle read by rows: T(n,k), n>=1, k>=1, in which column k lists k+1 copies of the squares in nondecreasing order, and the first element of column k is in row k(k+1)/2.


16



0, 0, 1, 0, 1, 0, 4, 0, 4, 1, 0, 9, 1, 0, 9, 1, 0, 16, 4, 0, 16, 4, 1, 0, 25, 4, 1, 0, 25, 9, 1, 0, 36, 9, 1, 0, 36, 9, 4, 0, 49, 16, 4, 1, 0, 49, 16, 4, 1, 0, 64, 16, 4, 1, 0, 64, 25, 9, 1, 0, 81, 25, 9, 1, 0, 81, 25, 9, 4, 0, 100, 36, 9, 4, 1, 0, 100, 36, 16, 4, 1, 0, 121, 36, 16, 4, 1, 0, 121, 49, 16, 4, 1, 0
(list;
graph;
refs;
listen;
history;
text;
internal format)



OFFSET

1,7


COMMENTS

Gives an identity for the sum of remainders of n mod k, for k = 1,2,3,...,n. Alternating sum of row n equals A004125(n), i.e., sum_{k=1..A003056(n))} (1)^(k1)*T(n,k) = A004125(n).
Row n has length A003056(n) hence the first element of column k is in row A000217(k).


LINKS



EXAMPLE

Triangle begins:
0;
0;
1, 0;
1, 0;
4, 0;
4, 1, 0;
9, 1, 0;
9, 1, 0;
16, 4, 0;
16, 4, 1, 0;
25, 4, 1, 0;
25, 9, 1, 0;
36, 9, 1, 0;
36, 9, 4, 0;
49, 16, 4, 1, 0;
49, 16, 4, 1, 0;
64, 16, 4, 1, 0;
64, 25, 9, 1, 0;
81, 25, 9, 1, 0;
81, 25, 9, 4, 0;
100, 36, 9, 4, 1, 0;
100, 36, 16, 4, 1, 0;
121, 36, 16, 4, 1, 0;
121, 49, 16, 4, 1, 0;
...
For n = 24 the 24th row of triangle is 121, 49, 16, 4, 1, 0 therefore the alternating row sum is 121  49 + 16  4 + 1  0 = 85 equaling A004125(24).


CROSSREFS

Cf. A000203, A000217, A000290, A003056, A004125, A120444, A196020, A211343, A228813, A231345, A231347, A235791, A235794, A236104, A236106, A237048, A237591, A237593, A261699.


KEYWORD

nonn,tabf


AUTHOR



STATUS

approved



