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 A351824 Irregular triangle read by rows: T(n,k) is the number of partitions of n into 2*k-1 consecutive parts, n >= 1, k >= 1. Column k lists 1's interleaved with 2*k-2 zeros, and the first element of column k is in row A000384(k). 6
 1, 1, 1, 1, 1, 1, 1, 1, 0, 1, 0, 1, 1, 1, 0, 1, 0, 1, 1, 1, 0, 1, 0, 1, 1, 1, 1, 0, 0, 1, 0, 0, 1, 1, 0, 1, 0, 0, 1, 0, 1, 1, 1, 0, 1, 0, 0, 1, 0, 0, 1, 1, 0, 1, 0, 1, 1, 0, 0, 1, 1, 0, 1, 0, 0, 1, 1, 0, 0, 0, 1, 1, 1, 0, 1, 0, 0, 0, 1, 0, 0, 0, 1, 1, 0, 0, 1, 0, 0, 0, 1, 0, 1, 1, 1, 1, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1 COMMENTS Conjecture 1: T(n,k) is the number of subparts, in an octant of the symmetric representation of sigma(n), that arises from the (2*k-1)-th double-staircase of the double-staircases diagram of n described in A335616. Conjecture 2: Indices of 1's coincide with indices of nonzero terms in A347263, A347529, A351819. For the above conjectures see also the "ziggurat" diagram described in A347186. This triangle is formed by the odd-indexed columns of the triangle A237048. Terms can be 0 or 1. LINKS Paolo Xausa, Table of n, a(n) for n = 1..10490 (rows 1..800 of triangle, flattened). FORMULA T(n,k) = A352499(n,k)/n. - Omar E. Pol, Mar 24 2022 T(n,k) = [(2*k-1)|n], where 1 <= k <= floor((sqrt(8*n+1)+1)/4) and [] is the Iverson bracket. - Paolo Xausa, Apr 01 2023 EXAMPLE Triangle begins: ----------------------- n / k 1 2 3 4 ----------------------- 1 | 1; 2 | 1; 3 | 1; 4 | 1; 5 | 1; 6 | 1, 1; 7 | 1, 0; 8 | 1, 0; 9 | 1, 1; 10 | 1, 0; 11 | 1, 0; 12 | 1, 1; 13 | 1, 0; 14 | 1, 0; 15 | 1, 1, 1; 16 | 1, 0, 0; 17 | 1, 0, 0; 18 | 1, 1, 0; 19 | 1, 0, 0; 20 | 1, 0, 1; 21 | 1, 1, 0; 22 | 1, 0, 0; 23 | 1, 0, 0; 24 | 1, 1, 0; 25 | 1, 0, 1; 26 | 1, 0, 0; 27 | 1, 1, 0; 28 | 1, 0, 0, 1; ... For n = 15 the partitions of 15 into an odd number of consecutive parts are [15], [6, 5, 4] and [5, 4, 3, 2, 1]. There are a partition with only one part, a partition with three parts and a partition with five parts, so the 15th row of triangle is [1, 1, 1]. MATHEMATICA A351824[rowmax_]:=Table[Boole[Divisible[n, 2k-1]], {n, rowmax}, {k, Floor[(Sqrt[8n+1]+1)/4]}]; A351824[50] (* Paolo Xausa, Apr 01 2023 *) CROSSREFS Row sums give A082647. Row n has length A351846(n). Cf. A000384, A196020, A235791, A236104, A237048, A237270, A237271, A237591, A237593, A244250, A262619, A262626, A279387, A280850, A280851, A286000, A286001, A296508, A299765, A335616, A347186, A347263, A347529, A348854, A351819, A352257, A352499. Sequence in context: A248863 A328306 A267256 * A365605 A365716 A334460 Adjacent sequences: A351821 A351822 A351823 * A351825 A351826 A351827 KEYWORD nonn,tabf,easy AUTHOR Omar E. Pol, Feb 20 2022 STATUS approved

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Last modified September 11 17:54 EDT 2024. Contains 375839 sequences. (Running on oeis4.)