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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.
1
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
1,2
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.
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.
EXAMPLE
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.
KEYWORD
nonn,tabf
AUTHOR
Omar E. Pol, Apr 05 2016
STATUS
approved