A306470 -id:A306470 - cp's OEIS frontend

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

Jan Koornstra, Aug 21 2019

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

			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 .

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

Cf. A309898, A003602, A002260, A306470.

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)

A309899 Numbers k = 1, 2, 3, ... are added to the sequence along with k blanks. Next, the numbers k + 1 are inserted on the blanks along with k + 1 blanks. This process is then repeated.

Original entry on oeis.org

1, 2, 2, 3, 4, 3, 3, 5, 6, 4, 4, 4, 7, 8, 5, 5, 9, 5, 5, 10, 6, 6, 11, 12, 6, 6, 6, 7, 7, 13, 14, 8, 7, 7, 15, 8, 7, 7, 16, 9, 17, 8, 8, 18, 9, 10, 8, 8, 8, 19, 20, 9, 9, 11, 10, 21, 22, 9, 9, 12, 9, 9, 10, 10, 23, 11, 24, 13, 25, 10, 10, 11, 26

Offset: 1

Jan Koornstra, Aug 21 2019

			The first elements are:
  1 _ 2 _ _ 3 _ _ _ 4 _ _ _ _ 5 _ _ _ _ _
    2   _ _   3 _ _   _ 4 _ _   _ _ 5 _ _
        3 _     _ _   4   _ _   _ _   5 _
          4     _ _       _ _   5 _     _
                5 _       _ _     _     _
                  .
  1 2 2 3 4 3 3 5 6 4 4 4 7 8 5 5 9 5 5 10

Jan Koornstra, Table of n, a(n) for n = 1..10000

Cf. A309898, A003602, A002260, A306470.

seq = []
for n in range(1, 100): seq += [n] + n*[-1]
for n in range(2, len(seq)):
  value = n
  gaps = 0
  position = n - 1
  while position < len(seq):
    if seq[position] == -1:
      if gaps > 0: gaps -= 1
      else:
        seq[position] = value
        gaps = value
        value += 1
    position += 1
print(seq)

cp's OEIS Frontend

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.

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