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, 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

Offset: 1

Author

Luc Rousseau, Feb 27 2019

Keywords

• nonn

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.

Examples

			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.
		

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)

Crossrefs

Cf. A177862, A072964, A069999.

Programs

Mathematica

T[n_] := n*(n + 1)/2
R[n_] := (Sqrt[8*n + 1] - 1)/2
S[0] := 0
S[d_] := S[d] =
  Module[{r = R[d]},
   If[IntegerQ[r], r++; r + T[r],
    First@TakeSmallest[
       1]@(S[#[[1]]] + S[#[[2]]] & /@ 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]]]