A271343 - OEIS

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A271343

Triangle read by rows: T(n,k) = A196020(n,k) - A266537(n,k), n>=1, k>=1.

1, 1, 5, 1, 1, 0, 9, 3, 1, -2, 1, 13, 5, 0, 1, 0, 0, 17, 7, 3, 1, -6, 0, 1, 21, 9, 0, 0, 1, 0, 3, 0, 25, 11, 0, 0, 1, -10, 0, 3, 29, 13, 7, 0, 1, 1, 0, 0, 0, 0, 33, 15, 0, 0, 0, 1, -14, 3, 5, 0, 37, 17, 0, 0, 0, 1, 0, 0, -2, 3, 41, 19, 11, 0, 0, 1, 1, -18, 0, 7, 0, 0, 45, 21, 0, 0, 0, 0, 1, 0, 3, 0, 0, 0

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OFFSET

1,3

COMMENTS

Gives an identity for A000593. Alternating sum of row n equals the sum of odd divisors of n, i.e., Sum_{k=1..A003056(n)} (-1)^(k-1)*T(n,k) = A000593(n).

Row n has length A003056(n) hence the column k starts in row A000217(k).

Since the odd-indexed rows of the triangle A266537 contain all zeros then odd-indexed rows of this triangle are the same as the odd-indexed rows of the triangle A196020.

If T(n,k) is the second odd number in the column k then T(n+1,k+1) = 1 is the first element in the column k+1.

Alternating row sums of A196020 give A000203.

Alternating row sums of A266537 give A146076.

LINKS

Table of n, a(n) for n=1..94.

EXAMPLE

Triangle begins:

5, 1;

1, 0;

9, 3;

1, -2, 1;

13, 5, 0;

1, 0, 0;

17, 7, 3;

1, -6, 0, 1;

21, 9, 0, 0;

1, 0, 3, 0;

25, 11, 0, 0;

1, -10, 0, 3;

29, 13, 7, 0, 1;

1, 0, 0, 0, 0;

33, 15, 0, 0, 0;

1, -14, 3, 5, 0;

37, 17, 0, 0, 0;

1, 0, 0, -2, 3;

41, 19, 11, 0, 0, 1;

1, -18, 0, 7, 0, 0;

45, 21, 0, 0, 0, 0;

1, 0, 3, 0, 0, 0;

49, 23, 0, 0, 5, 0;

1, -22, 0, 9, 0, 0;

53, 25, 15, 0, 0, 3;

1, 0, 0, -6, 0, 0, 1;

...

For n = 18 the divisors of 18 are 1, 2, 3, 6, 9, 18 and the sum of odd divisors of 18 is 1 + 3 + 9 = 13. On the other hand, the 18th row of the triangle is 1, -14, 3, 5, 0, so the alternating row sum is 1 -(-14) + 3 - 5 + 0 = 13, equaling the sum of odd divisors of 18.

CROSSREFS

Column 1 is A266072.

Cf. A000217, A000593, A001227, A003056, A146076, A196020, A236104, A237593, A261699, A266537.

Sequence in context: A220962 A201292 A348975 * A086464 A227577 A281446

Adjacent sequences: A271340 A271341 A271342 * A271344 A271345 A271346

KEYWORD

sign,tabf

AUTHOR

Omar E. Pol, Apr 06 2016

STATUS

approved