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A231347
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Triangle read by rows: T(n,k), n>=1, k>=1, in which column k lists the odd numbers interleaved with k-1 zeros but T(n,1) = n - 1 and the first element of column k is in row k(k+1)/2.
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13
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0, 1, 2, 1, 3, 0, 4, 3, 5, 0, 1, 6, 5, 0, 7, 0, 0, 8, 7, 3, 9, 0, 0, 1, 10, 9, 0, 0, 11, 0, 5, 0, 12, 11, 0, 0, 13, 0, 0, 3, 14, 13, 7, 0, 1, 15, 0, 0, 0, 0, 16, 15, 0, 0, 0, 17, 0, 9, 5, 0, 18, 17, 0, 0, 0, 19, 0, 0, 0, 3, 20, 19, 11, 0, 0, 1, 21, 0, 0, 7, 0, 0
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OFFSET
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1,3
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COMMENTS
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Alternating sum of row n equals the sum of aliquot divisors of n, i.e., sum_{k=1..A003056(n)} (-1)^(k-1)*T(n,k) = A001065(n).
The number of positive terms in row n is A001227(n), for n >= 2.
If n = 2^j then the only positive integer in row n is T(n,1) = n - 1, for j >= 1.
If n is an odd prime then the only two positive integers in row n are T(n,1) = n - 1 and T(n,2) = n - 2.
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LINKS
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FORMULA
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T(n,1) = n - 1.
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EXAMPLE
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Triangle begins:
0;
1;
2, 1;
3, 0;
4, 3;
5, 0, 1;
6, 5, 0;
7, 0, 0;
8, 7, 3;
9, 0, 0, 1;
10, 9, 0, 0;
11, 0, 5, 0;
12, 11, 0, 0;
13, 0, 0, 3;
14, 13, 7, 0, 1;
15, 0, 0, 0, 0;
16, 15, 0, 0, 0;
17, 0, 9, 5, 0;
18, 17, 0, 0, 0;
19, 0, 0, 0, 3;
20, 19, 11, 0, 0, 1;
21, 0, 0, 7, 0, 0;
22, 21, 0, 0, 0, 0;
23, 0, 13, 0, 0, 0;
...
For n = 15 the aliquot divisors of 15 are 1, 3, 5, therefore the sum of aliquot divisors of 15 is 1 + 3 + 5 = 9. On the other hand the 15th row of triangle is 14, 13, 7, 0, 1, hence the alternating row sum is 14 - 13 + 7 - 0 + 1 = 9, equalling the sum of aliquot divisors of 15.
If n is even then the alternating sum of the n-th row of triangle is simpler than the sum of aliquot divisors of n. Example: the sum of aliquot divisors of 24 is 1 + 2 + 3 + 4 + 6 + 8 + 12 = 36, and the alternating sum of the 24th row of triangle is 23 - 0 + 13 - 0 + 0 - 0 = 36.
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CROSSREFS
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Cf. A000217, A001065, A001227, A000203, A003056, A196020, A211343, A212119, A228813, A231345, A235791, A235794, A236104, A236106, A236112, A237591, A237593, A286001.
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KEYWORD
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nonn,tabf
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AUTHOR
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STATUS
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approved
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