|
|
A236106
|
|
Triangle read by rows: T(n,k), n>=1, k>=1, in which column k lists the twice odd numbers (A016825) interleaved with k-1 zeros, and the first element of column k is in row k(k+1)/2.
|
|
20
|
|
|
2, 6, 10, 2, 14, 0, 18, 6, 22, 0, 2, 26, 10, 0, 30, 0, 0, 34, 14, 6, 38, 0, 0, 2, 42, 18, 0, 0, 46, 0, 10, 0, 50, 22, 0, 0, 54, 0, 0, 6, 58, 26, 14, 0, 2, 62, 0, 0, 0, 0, 66, 30, 0, 0, 0, 70, 0, 18, 10, 0, 74, 34, 0, 0, 0, 78, 0, 0, 0, 6, 82, 38, 22, 0, 0, 2
(list;
graph;
refs;
listen;
history;
text;
internal format)
|
|
|
OFFSET
|
1,1
|
|
COMMENTS
|
Gives an identity for the twice sigma function (A074400), the sum of the even divisors of 2n.
Row n has length A003056(n) hence the first element of column k is in row A000217(k).
The number of positive terms in row n is A001227(n).
|
|
LINKS
|
|
|
FORMULA
|
|
|
EXAMPLE
|
Triangle begins:
2;
6;
10, 2;
14, 0;
18, 6;
22, 0, 2;
26, 10, 0;
30, 0, 0;
34, 14, 6;
38, 0, 0, 2;
42, 18, 0, 0;
46, 0, 10, 0;
50, 22, 0, 0;
54, 0, 0, 6;
58, 26, 14, 0, 2;
62, 0, 0, 0, 0;
66, 30, 0, 0, 0;
70, 0, 18, 10, 0;
74, 34, 0, 0, 0;
78, 0, 0, 0, 6;
82, 38, 22, 0, 0, 2;
86, 0, 0, 14, 0, 0;
90, 42, 0, 0, 0, 0;
94, 0, 26, 0, 0, 0;
...
For n = 9 the divisors of 2*9 = 18 are 1, 2, 3, 6, 9, 18, therefore the sum of the even divisors of 18 is 2 + 6 + 18 = 26. On the other hand the 9th row of triangle is 34, 14, 6, therefore the alternating row sum is 34 - 14 + 6 = 26, equaling the sum of the even divisors of 18.
If n is even then the alternating sum of the n-th row of triangle is simpler than the sum of the even divisors of 2n. Example: for n = 12 the sum of the even divisors of 2*12 = 24 is 2 + 4 + 6 + 8 + 12 + 24 = 56, and the alternating sum of the 12th row of triangle is 46 - 0 + 10 - 0 = 56.
|
|
CROSSREFS
|
Cf. A000203, A000217, A001227, A003056, A016825, A074400, A196020, A211343, A228813, A231345, A231347, A235791, A235794, A236104, A236112.
|
|
KEYWORD
|
nonn,tabf
|
|
AUTHOR
|
|
|
STATUS
|
approved
|
|
|
|