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 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.). 2
 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 (list; graph; refs; listen; history; text; internal format)
 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 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: A337939 A319020 A099200 * A172398 A070107 A299908 Adjacent sequences:  A093575 A093576 A093577 * A093579 A093580 A093581 KEYWORD easy,nonn AUTHOR Howard A. Landman, Apr 01 2004 STATUS approved

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Last modified June 23 11:02 EDT 2021. Contains 345397 sequences. (Running on oeis4.)