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.

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

Offset: 1

Author

Omar E. Pol, Jan 23 2014

Keywords

• nonn
• tabf

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)^(k-1)*T(n,k) = A004125(n).

Row n has length A003056(n) hence the first element of column k is in row A000217(k).

Examples

			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.