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 A306597 a(n) = Card({ Sum_{k=1..n}(x_k * k) : (x_k)_{k=1..n} is an n-tuple of nonnegative integers such that Sum_{k=1..n}(x_k * T_k) = T_n }), where T_k denotes the k-th triangular number. 1
 1, 2, 4, 6, 9, 15, 20, 27, 34, 43, 52, 63, 75, 87, 102, 117, 132, 149, 166, 185, 206, 226, 248, 271, 294, 318, 345, 373, 399, 429, 459, 489, 520, 554, 587, 623, 658, 695, 734, 772, 811, 853, 894, 936, 981, 1026, 1072, 1119, 1167, 1215, 1266, 1316, 1368, 1420 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,2 COMMENTS Inspired by the questions: - Q1: into how many regions do n+1 straight lines divide the plane? - Q2: what is the number of possible answers to Q1? This sequence provides an answer to an analog of Q2 in a modified version of the problem. Also an analog of A069999(n) with the roles of k and T_k swapped in the definition. LINKS Luc Rousseau, Table of n, a(n) for n = 1..150 Luc Rousseau, C program Luc Rousseau, Java program (standalone version) Luc Rousseau, Java program (jOEIS version, github) EXAMPLE When n = 3, n*(n+1)/2 = 6. All possible ways to partition 6 into parts with triangular sizes (1, 3, 6) are:   0*1 + 0*3 + 1*6 = 6   0*1 + 2*3 + 0*6 = 6   3*1 + 1*3 + 0*6 = 6   6*1 + 0*3 + 0*6 = 6 In the above products, keep the left multiplicands and replace the right ones with their triangular roots:   0*1 + 0*2 + 1*3 = 3   0*1 + 2*2 + 0*3 = 4   3*1 + 1*2 + 0*3 = 5   6*1 + 0*2 + 0*3 = 6 Card({ 3, 4, 5, 6 }) = 4, so a(3) = 4. MATHEMATICA T[n_] := n*(n + 1)/2 R[n_] := (Sqrt[8*n + 1] - 1)/2 S := 0 S[d_] := S[d] =   Module[{r = R[d]},    If[IntegerQ[r], r++; r + T[r],     First@TakeSmallest[        1]@(S[#[]] + S[#[]] & /@ IntegerPartitions[d, {2}])]] A0[n_] := Sum[Boole[d + S[d] <= 2*n], {d, 0, n}] A[n_] := A0[T[n]] For[n = 1, n <= 150, n++, Print[n, " ", A[n]]] CROSSREFS Cf. A177862, A072964, A069999. Sequence in context: A283024 A090483 A299494 * A127740 A323432 A327046 Adjacent sequences:  A306594 A306595 A306596 * A306598 A306599 A306600 KEYWORD nonn AUTHOR Luc Rousseau, Feb 27 2019 STATUS approved

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Last modified June 19 14:40 EDT 2021. Contains 345140 sequences. (Running on oeis4.)