login
The OEIS Foundation is supported by donations from users of the OEIS and by a grant from the Simons Foundation.

 

Logo


Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
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] := 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]]]

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

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recent
The OEIS Community | Maintained by The OEIS Foundation Inc.

License Agreements, Terms of Use, Privacy Policy. .

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