This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A093578 #6 Feb 05 2025 22:23:21 %S A093578 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, %T A093578 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, %U A093578 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 %N 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.). %C A093578 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, so that a(A093579(n)) > 0 for all n. %e A093578 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. %Y A093578 Cf. A000217, A093579. %K A093578 easy,nonn %O A093578 0,29 %A A093578 _Howard A. Landman_, Apr 01 2004