A093578 - OEIS

A093578

Take n sheets of paper, arrange them into piles, write on each sheet the cardinality (number of sheets) of its pile. Do this again, so each sheet is labeled by an ordered pair of positive integers. How many ways can this be done so that every sheet has a unique label? (Only distinct sets of labels count, not every permutation of the labels or sheets.).

1, 1, 0, 1, 1, 0, 1, 1, 0, 0, 1, 1, 0, 1, 1, 1, 1, 0, 1, 1, 0, 1, 1, 0, 1, 1, 0, 1, 2, 1, 0, 1, 1, 0, 1, 1, 1, 1, 0, 1, 1, 0, 1, 1, 0, 1, 2, 1, 1, 2, 1, 1, 1, 0, 0, 2, 2, 0, 2, 2, 0, 1, 1, 0, 0, 1, 2, 1, 1, 2, 2, 1, 1, 2, 1, 0, 1, 1, 1, 2, 1, 2, 2, 0, 2, 2, 0, 0, 1, 1, 0, 2, 2, 1, 2, 1, 1, 2, 1, 1, 1, 1, 2, 1, 1

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OFFSET

0,29

COMMENTS

If n is a triangular number (A000217), then there is a trivial solution using piles of 1,2,3,...,k, where n = k(k+1)/2. All solutions are based on sums of triangular numbers, but not all such sums are legal. No indices of the triangular numbers can have a ratio smaller than 2; if they do then labels from the two triangles are not disjoint. a(28) = 2 because we can either use the trivial T(7) = 28 solution or the T(6) + T(3) + T(1) = 21 + 6 + 1 = 28 solution. A093579 gives the integers for which there is a solution and A093580 those for which there is no solution, so that a(A093579(n)) > 0 and a(A093580(n)) = 0 for all n.

LINKS

Table of n, a(n) for n=0..104.

EXAMPLE

a(1) = 1 because the only possible label is (1,1); a(2) = 0 because there is no way to prevent both pieces of paper from getting labeled identically.

CROSSREFS

Cf. A000217, A093579, A093580.

Sequence in context: A319020 A099200 A358351 * A172398 A070107 A299908

Adjacent sequences: A093575 A093576 A093577 * A093579 A093580 A093581

KEYWORD

easy,nonn

AUTHOR

Howard A. Landman, Apr 01 2004

STATUS

approved