A309898 For each b = 1, 2, 3, ... numbers k = 1, 2, 3, ... are inserted into the blanks within the sequence along with k * b blanks, skipping existing terms in the sequence.

1, 1, 2, 1, 1, 3, 2, 1, 1, 4, 2, 1, 3, 1, 5, 2, 1, 1, 2, 3, 6, 4, 1, 1, 2, 1, 1, 7, 3, 2, 1, 5, 1, 4, 2, 8, 1, 3, 1, 2, 1, 1, 2, 6, 9, 3, 4, 1, 1, 5, 2, 1, 1, 3, 10, 2, 1, 1, 7, 4, 2, 1, 3, 1, 2, 11, 1, 5, 6, 1, 2, 3, 4, 1, 8, 1, 2, 12, 1, 1, 3, 2, 1, 1, 4, 5, 2, 1, 3, 7, 13

Offset: 1

Author

Jan Koornstra, Aug 21 2019

Keywords

• nonn
• easy

Comments

b = 1 for all blanks forms A003602. b = k yields A002260.

Examples

			The first 21 terms are constructed as follows:
  1 _ 2 _ _ 3 _ _ _ 4 _ _ _ _ 5 _ _ _ _ _ .
    1   _ _   2 _ _   _ _ 3 _   _ _ _ _ _ .
        1 _     _ _   2 _   _   _ _ _ _ 3 .
          1     _ _     _   _   2 _ _ _   .
                1 _     _   _     _ _ 2   .
                  1     _   _     _ _     .
                        1   _     _ _     .
                            1     _ _     .
                                  1 _     .
                                    1     .
  1 1 2 1 1 3 2 1 1 4 2 1 3 1 5 2 1 1 2 3 .
		

Links

• Seiichi Manyama, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A309898, A003602, A002260, A306470.

Programs

Python

seq = []
b = 2
for n in range(1, 100):
  seq += [n] + [-1] * n
while -1 in seq:
  i = seq.index(-1)
  seq[i] = 1
  k = 2
  blanks = b
  for s in range(i + 1, len(seq)):
    if seq[s] == -1:
      blanks -= 1
      if blanks < 0:
        seq[s] = k
        blanks = k * b
        k += 1
  b += 1
print(seq)