A292510 a(n) = smallest k >= 1 such that {1, p(n,2), p(n,3), ..., p(n,k)} can be partitioned into two sets with equal sums, where p(n,m) is m-th n-gonal number.

4, 7, 6, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7

Offset

3,1

Comments

Conjecture: a(n) = 7 for n > 5.

Links

Table of n, a(n) for n=3..90.

Wikipedia, Polygonal number

Formula

p(n,1) + p(n,2) + p(n,4) + p(n,7) = p(n,3) + p(n,5) + p(n,6) (= 28*n-42). So a(n) <= 7.

Example

n = 3
1+3+6 = 10
n = 4
1+4+16+49 = 9+25+36 (= 70 = 28*4-42)
n = 5
1+5+22+35 = 12+51 (=63)
n = 6
1+6+28+91 = 15+45+66 (= 126 = 28*6-42)

Prog

(Ruby)
def f(k, n)
  n * ((k - 2) * n - k + 4) / 2
end
def A(k, n)
  ary = [1]
  s_ary = [0]
  (1..n).each{|i| s_ary << s_ary[-1] + f(k, i)}
  m = s_ary[-1]
  a = Array.new(m + 1){0}
  a[0] = 1
  (1..n).each{|i|
    b = a.clone
    (0..[s_ary[i - 1], m - f(k, i)].min).each{|j| b[j + f(k, i)] += a[j]}
    a = b
    s_ary[i] % 2 == 0 ? ary << a[s_ary[i] / 2] : ary << 0
  }
  ary
end
def B(n)
  i = 1
  while A(n, i)[-1] == 0
    i += 1
  end
  i
end
def A292510(n)
  (3..n).map{|i| B(i)}
end
p A292510(100)

Crossrefs

Cf. A019568, A158092, A158380, A292474.

Sequence in context: A275162 A105228 A199550 * A284116 A200353 A081845

Adjacent sequences: A292507 A292508 A292509 * A292511 A292512 A292513

Keyword

nonn

Author

Seiichi Manyama, Sep 17 2017

Status

approved