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A231345 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) = -1 and the first element of column k is in row k(k+1)/2. 12
-1, -1, -1, 1, -1, 0, -1, 3, -1, 0, 1, -1, 5, 0, -1, 0, 0, -1, 7, 3, -1, 0, 0, 1, -1, 9, 0, 0, -1, 0, 5, 0, -1, 11, 0, 0, -1, 0, 0, 3, -1, 13, 7, 0, 1, -1, 0, 0, 0, 0, -1, 15, 0, 0, 0, -1, 0, 9, 5, 0, -1, 17, 0, 0, 0, -1, 0, 0, 0, 3, -1, 19, 11, 0, 0, 1 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,8

COMMENTS

Gives an identity for the abundance of n. Alternating sum of row n equals the abundance of n, i.e., sum_{k=1..A003056(n))} (-1)^(k-1)*T(n,k) = A033880(n).

Row n has length A003056(n) hence the first element of column k is in row A000217(k).

LINKS

Table of n, a(n) for n=1..76.

Eric Weisstein's World of Mathematics, Abundance

Eric Weisstein's World of Mathematics, Quasiperfect Number

FORMULA

T(n,1) = -1; T(n,k) = A196020(n,k), for k >= 2.

EXAMPLE

Triangle begins:

  -1;

  -1;

  -1,  1;

  -1,  0;

  -1,  3;

  -1,  0,  1;

  -1,  5,  0;

  -1,  0,  0;

  -1,  7,  3;

  -1,  0,  0,  1;

  -1,  9,  0,  0;

  -1,  0,  5,  0;

  -1, 11,  0,  0;

  -1,  0,  0,  3;

  -1, 13,  7,  0,  1;

  -1,  0,  0,  0,  0;

  -1, 15,  0,  0,  0;

  -1,  0,  9,  5,  0;

  -1, 17,  0,  0,  0;

  -1,  0,  0,  0,  3;

  -1, 19, 11,  0,  0,  1;

  -1,  0,  0,  7,  0,  0;

  -1, 21,  0,  0,  0,  0;

  -1,  0, 13,  0,  0,  0;

  ...

For n = 15 the divisors of 15 are 1, 3, 5, 15 hence the abundance of 15 is 1 + 3 + 5 + 15 - 2*15 = 1 + 3 + 5 - 15 = -6. On the other hand the 15th row of triangle is -1, 13, 7, 0, 1, hence the alternating row sum is -1 - 13 + 7 - 0 + 1 = -6, equalling the abundance of 15.

If n is even then the alternating sum of the n-th row of triangle is simpler than the sum of divisors of n minus 2*n. Example: the sum of divisors of 24 minus 2*24 is 1 + 2 + 3 + 4 + 6 + 8 + 12 + 24 - 2*24 = 60 - 48 = 12, and the alternating sum of the 24th row of triangle is -1 - 0 + 13 - 0 + 0 - 0 = 12.

CROSSREFS

Column 2 is A193356.

Cf. A000203, A000217, A000396, A003056, A005100, A005843, A033879, A033880, A069283, A196020, A212119, A228813, A231347, A235791, A235794, A236104, A236106, A236112, A237593.

Sequence in context: A250484 A294250 A117417 * A271344 A183134 A053382

Adjacent sequences:  A231342 A231343 A231344 * A231346 A231347 A231348

KEYWORD

sign,tabf,nice

AUTHOR

Omar E. Pol, Dec 26 2013

STATUS

approved

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Last modified February 21 04:18 EST 2019. Contains 320371 sequences. (Running on oeis4.)