A293866 -id:A293866 - cp's OEIS frontend

A344264 When computing A229037, the value of the n-th term will add up to n-1 new constraints on the n-1 following terms; a(n) is the number of new constraints that the computation of A229037(n) brings.

0, 1, 2, 2, 2, 5, 5, 7, 7, 4, 4, 8, 3, 3, 9, 9, 16, 15, 9, 11, 12, 15, 15, 23, 12, 14, 23, 8, 7, 16, 6, 6, 15, 14, 26, 21, 5, 6, 10, 4, 4, 12, 11, 27, 24, 13, 19, 18, 34, 31, 49, 24, 28, 47, 46, 22, 19, 21, 23, 48, 18, 18, 44, 20, 39, 57, 47, 40, 38, 43, 46

Offset: 1

Rémy Sigrist, May 13 2021

nonn

Once A229037(n) has been computed, we have the following constraints:
- for k = 1..n-1, A229037(n + k) <> 2*A229037(n) - A229037(n - k),
   - if 2*A229037(n) - A229037(n - k) <= 0, then we ignore this constraint,
   - if 2*A229037(n) - A229037(n - k) = 2*A229037(n') - A229037(n' - k')
     for some n' < n and k' < n' such that n + k = n' + k',
     then we also ignore this constraint.

			The first terms, alongside the corresponding value of A229037 (denoted by f(n)) and the new constraints, are:
  n  a(n)  f(n)  1  2  3  4  5  6  7  8  9  10  11  12  13  14  15  16  17  18  19
  -  ----  ----  -  -  -  -  -  -  -  -  -  --  --  --  --  --  --  --  --  --  --
  1     0     1  .  .  .  .  .  .  .  .  .  .   .   .   .   .   .   .   .   .   .
  2     1     1  .  .  1  .  .  .  .  .  .  .   .   .   .   .   .   .   .   .   .
  3     2     2  .  .  .  3  3  .  .  .  .  .   .   .   .   .   .   .   .   .   .
  4     2     1  .  .  .  .  .  1  1  .  .  .   .   .   .   .   .   .   .   .   .
  5     2     1  .  .  .  .  .  .  .  1  1  .   .   .   .   .   .   .   .   .   .
  6     5     2  .  .  .  .  .  .  3  3  2   3   3  .   .   .   .   .   .   .   .
  7     5     2  .  .  .  .  .  .  .  2  3  .    2   3   3  .   .   .   .   .   .
  8     7     4  .  .  .  .  .  .  .  .  6   6   7   7   6   7   7  .   .   .   .
  9     7     4  .  .  .  .  .  .  .  .  .   4   6   6   7  .    6   7   7  .   .

Rémy Sigrist, Table of n, a(n) for n = 1..10000
Rémy Sigrist, Scatterplot of the first 100000 terms
Rémy Sigrist, PARI program for A344264

Cf. A229037, A293866.

PARI
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See Links section.
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a(n) < n.

A344322 a(n) is the number of forbidden values when computing A229037(n).

Original entry on oeis.org

0, 0, 1, 1, 1, 1, 2, 3, 4, 3, 4, 4, 3, 2, 3, 3, 4, 4, 4, 5, 5, 6, 6, 7, 8, 7, 9, 9, 8, 9, 9, 9, 8, 9, 9, 8, 8, 8, 8, 7, 8, 6, 8, 9, 7, 8, 9, 7, 9, 8, 9, 9, 8, 9, 10, 10, 11, 10, 11, 11, 11, 11, 11, 11, 13, 15, 14, 14, 13, 15, 17, 16, 17, 15, 16, 17, 16, 16, 17

Offset: 1

Rémy Sigrist, May 15 2021

nonn

In other words, a(n) is the number of distinct positive terms of the form 2*A229037(n - k) - A229037(n - 2*k) with n > 2*k > 0.

			For n=14, we have:
- 2*A229037(13) - A229037(12) = 2*1 - 2 = 0,
- 2*A229037(12) - A229037(10) = 2*2 - 1 = 3,
- 2*A229037(11) - A229037(8) = 2*1 - 4 = -2,
- 2*A229037(10) - A229037(6) = 2*1 - 2 = 0,
- 2*A229037(9) - A229037(4) = 2*4 - 1 = 7,
- 2*A229037(8) - A229037(2) = 2*4 - 1 = 7,
- so a(14) = #{3, 7} = 2.

Rémy Sigrist, Table of n, a(n) for n = 1..10000
Rémy Sigrist, PARI program for A344322

Cf. A229037, A293866, A344264.

PARI
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\\ See Links section.
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cp's OEIS Frontend

A344264 When computing A229037, the value of the n-th term will add up to n-1 new constraints on the n-1 following terms; a(n) is the number of new constraints that the computation of A229037(n) brings.

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