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 A331553 Irregular triangle T(n,k) = A115722(n,k)^2 - n. 0
 0, 0, 1, 1, 2, 2, 2, 3, 3, 0, 3, 3, 4, 4, 1, 4, 1, 4, 4, 5, 5, 2, 5, 2, 2, 5, 2, 2, 5, 5, 6, 6, 3, 6, 3, 3, 6, 3, 3, 3, 6, 3, 3, 6, 6, 7, 7, 4, 7, 4, 4, 7, 4, 4, 4, 4, 7, 4, 4, 4, 4, 7, 4, 4, 4, 7, 7, 8, 8, 5, 8, 5, 5, 8, 5, 5, 5, 5, 8, 5, 5, 5, 5, 5, 8, 0, 5 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,5 COMMENTS Let P be an integer partition of n, and let D be the Durfee square of P with side length s, thus area s^2. We borrow the term "square excess" from A053186(n), which is simply the difference n - floor(sqrt(n)). This sequence lists the "Durfee square excess" of P = n - s^2 for all partitions P of n in reverse lexicographic order. Zero appears in row n for n that are perfect squares. Let r = sqrt(n). For perfect square n, there exists a partition of n that consists of a run of r parts that are each r themselves; e.g., for n = 4, we have {2, 2}, for n = 9, we have {3, 3, 3}. It is clear through the Ferrers diagram of these partitions that they are equivalent to their Durfee square, thus n - s^2 = 0. Since the partitions of any n contain Durfee squares in the range of 1 <= s <= floor(sqrt(n)) (with perfect square n also including k = 0), the distinct Durfee square excesses must be the differences n - s^2 for 1 <= s <= floor(sqrt(n)). LINKS Table of n, a(n) for n=0..86. Eric Weisstein's World of Mathematics, Durfee Square. FORMULA T(n,k) = A115722(n,k)^2 - n. 2 * A116365(n) = sum of row n. EXAMPLE Table begins: 0: 0; 1: 0; 2: 1, 1; 3: 2, 2, 2; 4: 3, 3, 0, 3, 3; 5: 4, 4, 1, 4, 1, 4, 4; 6: 5, 5, 2, 5, 2, 2, 5, 2, 2, 5, 5; 7: 6, 6, 3, 6, 3, 3, 6, 3, 3, 3, 6, 3, 3, 6, 6; ... Table of distinct terms: 1: 0; 2: 1; 3: 2; 4: 0, 3; 5: 1, 4; 6: 2, 5; 7: 3, 6; 8: 4, 7; 9: 0, 5, 8; ... For n = 4, the partitions are {4}, {3, 1}, {2, 2}, {2, 1, 1}, {1, 1, 1, 1}. The partition {2, 2} has Durfee square s = 2; for all partitions except {2, 2}, we have Durfee square with s = 1. MATHEMATICA {0}~Join~Array[Map[Total@ # - Block[{k = Length@ #}, While[Nand[k > 0, AllTrue[Take[#, k], # >= k &]], k--]; k]^2 &, IntegerPartitions[#]] &, 12] // Flatten CROSSREFS Cf. A115722, A116365. Sequence in context: A219281 A125600 A084053 * A182663 A071452 A282495 Adjacent sequences: A331550 A331551 A331552 * A331554 A331555 A331556 KEYWORD nonn,tabf AUTHOR Michael De Vlieger, Jan 20 2020 STATUS approved

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Last modified March 3 00:46 EST 2024. Contains 370499 sequences. (Running on oeis4.)