A266537 Triangle read by rows: T(n,k), n>=1, k>=1, in which column k lists the twice odd numbers (A016825) interleaved with 2*k-1 zeros, and the first positive element of column k is in the row A002378(k), with T(1,1) = 0.

0, 2, 0, 6, 0, 10, 2, 0, 0, 14, 0, 0, 0, 18, 6, 0, 0, 22, 0, 2, 0, 0, 0, 26, 10, 0, 0, 0, 0, 30, 0, 0, 0, 0, 0, 34, 14, 6, 0, 0, 0, 38, 0, 0, 2, 0, 0, 0, 0, 42, 18, 0, 0, 0, 0, 0, 0, 46, 0, 10, 0, 0, 0, 0, 0, 50, 22, 0, 0, 0, 0, 0, 0, 54, 0, 0, 6, 0, 0, 0, 0, 58, 26, 14, 0, 2

Offset: 1

Author

Omar E. Pol, Apr 05 2016

Keywords

• nonn
• tabf

Comments

Gives an identity for A146076. Alternating sum in row n equals the sum of even divisors of n.

Even-indexed rows of the triangle give A236106.

If T(n,k) = 6 then T(n+2,k+1) = 2, the first element of the column k+1.

Examples

			Triangle begins:
0;
2;
0;
6;
0;
10,  2;
0,   0;
14,  0;
0,   0;
18,  6;
0,   0;
22,  0,  2;
0,   0,  0;
26, 10,  0;
0,   0,  0;
30,  0,  0;
0,   0,  0;
34, 14,  6;
0,   0,  0;
38,  0,  0,  2;
0,   0,  0,  0;
42, 18,  0,  0;
0,   0,  0,  0;
46,  0, 10,  0;
0,   0,  0,  0;
50, 22,  0,  0;
0,   0,  0,  0;
54,  0,  0,  6;
0,   0,  0,  0;
58, 26, 14,  0,  2;
...
For n = 12 the divisors of 12 are 1, 2, 3, 4, 6, 12 and the sum of even divisors of 12 is 2 + 4 + 6 + 12 = 24. On the other hand, the 12th row of the triangle is 22, 0, 2, so the alternating row sum is 22 - 0 + 2 = 24, equaling the sum of even divisors of 12.
		

Crossrefs

Cf. A002378, A016825, A146076, A196020, A236106, A237593, A271343.

Formula

T(n,k) = 0, if n is odd.

T(n,k) = 2*A196020(n/2,k) = A236106(n/2,k), if n is even.