A334474 -id:A334474 - cp's OEIS frontend

A334475 a(n) is the X-coordinate of the n-th point of the space filling curve T defined in A334474; sequence A334476 gives Y-coordinates.

0, 0, 1, 1, 0, 0, 1, 2, 2, 3, 3, 2, 3, 3, 2, 1, 0, 0, 1, 1, 0, 0, 1, 2, 2, 3, 4, 4, 5, 5, 4, 4, 5, 6, 6, 7, 7, 6, 7, 7, 6, 5, 4, 5, 4, 4, 5, 6, 7, 7, 6, 6, 7, 7, 6, 5, 5, 4, 3, 2, 3, 3, 2, 1, 0, 0, 1, 1, 0, 0, 1, 2, 2, 3, 3, 2, 3, 3, 2, 1, 0, 0, 1, 1, 0, 0, 1

Offset: 0

Rémy Sigrist, May 02 2020

nonn

			The space filling curve T starts as follows:
    5|  ...
     |   |
     |   |
    4|  16
     |   \
     |    \
    3|   5 15-14-13
     |   |\       |
     |   | \      |
    2|   4  6 11-12
     |    \  \  \
     |     \  \  \
    1|   1  3  7 10
     |   |\ |  |  |
     |   | \|  |  |
    0|   0  2  8--9
  ---+-------------
  y/x|   0  1  2  3
- hence a(2) = a(3) = a(6) = a(15) = 1.

Rémy Sigrist, Table of n, a(n) for n = 0..8255
Hugo Pfoertner, Illustration of curve T, using Plot 2.
Rémy Sigrist, PARI program for A334475
Index entries for sequences related to coordinates of 2D curves

Cf. A334474, A334476.

PARI
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See Links section.
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A334474(A334476(n), a(n)) = n.

A334476 a(n) is the Y-coordinate of the n-th point of the space filling curve T defined in A334474; sequence A334475 gives X-coordinates.

Original entry on oeis.org

0, 1, 0, 1, 2, 3, 2, 1, 0, 0, 1, 2, 2, 3, 3, 3, 4, 5, 4, 5, 6, 7, 6, 5, 4, 4, 3, 2, 2, 1, 1, 0, 0, 0, 1, 0, 1, 2, 2, 3, 3, 3, 4, 4, 5, 6, 5, 4, 4, 5, 5, 6, 6, 7, 7, 7, 6, 7, 7, 7, 6, 5, 6, 7, 8, 9, 8, 9, 10, 11, 10, 9, 8, 8, 9, 10, 10, 11, 11, 11, 12, 13, 12

Offset: 0

Rémy Sigrist, May 02 2020

			The space filling curve T starts as follows:
    5|  ...
     |   |
     |   |
    4|  16
     |   \
     |    \
    3|   5 15-14-13
     |   |\       |
     |   | \      |
    2|   4  6 11-12
     |    \  \  \
     |     \  \  \
    1|   1  3  7 10
     |   |\ |  |  |
     |   | \|  |  |
    0|   0  2  8--9
  ---+-------------
  y/x|   0  1  2  3
- hence a(1) = a(3) = a(7) = a(10) = 1.

Rémy Sigrist, Table of n, a(n) for n = 0..8255
Rémy Sigrist, PARI program for A334476
Index entries for sequences related to coordinates of 2D curves

Cf. A334474, A334475.

PARI
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See Links section.
```

A334474(a(n), A334475(n)) = n.

A342224 The n-th and a(n)-th points of the curve (A334474, A334475) are symmetrical with respect to the line X=Y.

Original entry on oeis.org

0, 2, 1, 3, 8, 9, 7, 6, 4, 5, 15, 11, 14, 13, 12, 10, 31, 32, 30, 29, 33, 35, 34, 28, 27, 26, 25, 24, 23, 19, 18, 16, 17, 20, 22, 21, 63, 62, 59, 58, 60, 61, 42, 44, 43, 47, 46, 45, 57, 55, 56, 51, 54, 53, 52, 49, 50, 48, 39, 38, 40, 41, 37, 36, 121, 122, 120

Offset: 0

Rémy Sigrist, Mar 06 2021

nonn

In other words, a(n) is the unique k such that A334474(n) = A334475(k) and A334474(k) = A334475(n).

This sequence is a self-inverse permutation of the nonnegative integers.

			The curve (A334474, A334475) begins as follows on a hexagonal lattice:
                 +
                /5\
               /   \
              +4    +6
               \     \
                \     \
           +    3+     +7
          /1\   /     /
         /   \ /     /
        +     +     +----+
         0     2     8    9
- so a(0) = 0,
     a(1) = 2,
     a(3) = 3,
     a(4) = 8,
     a(5) = 9,
     a(6) = 7.

Rémy Sigrist, Table of n, a(n) for n = 0..8255
Rémy Sigrist, PARI program for A342224
Index entries for sequences that are permutations of the natural numbers

See A342217 and A342218 for similar sequences.

Cf. A007582, A334474, A334475.

PARI
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See Links section.
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a(n) < A007582(k) for any n < A007582(k).

A344634 Numbers k such that half the numbers from 0 to k inclusive contain the digit "0".

Original entry on oeis.org

1, 10761677, 14958585, 14960717, 14961735, 15013205, 15588833, 15590573, 15591959, 15591961, 15592031, 15592229, 15592231, 15603695, 15633495, 15633503, 15633517, 16076087, 16263743, 20327615

Offset: 1

Glen Gilchrist, May 25 2021

Andrew Hilton (see Ref.) refers to these as "half-zero" numbers.

			1 is a term since among the numbers 0,1 exactly half contain a digit "0".
10761677 is a term since among the numbers 0,1,2,...,10761677 exactly half contain a digit "0".

Andrew Hilton, 101 Puzzles to Solve on your Microcomputer, 1984, HARRAP, page 57.

Cf. A016189, A334474, A344636.

def afind(limit):
  count0 = [0, 1]
  for k in range(1, limit+1):
    count0['0' in str(k)] += 1
    if count0[0] == count0[1]: print(k, end=", ")
afind(3*10**7) # Michael S. Branicky, May 25 2021

cp's OEIS Frontend

A334475 a(n) is the X-coordinate of the n-th point of the space filling curve T defined in A334474; sequence A334476 gives Y-coordinates.

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A334476 a(n) is the Y-coordinate of the n-th point of the space filling curve T defined in A334474; sequence A334475 gives X-coordinates.

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PARI

Formula

A342224 The n-th and a(n)-th points of the curve (A334474, A334475) are symmetrical with respect to the line X=Y.

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Examples

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Programs

PARI

Formula

A344634 Numbers k such that half the numbers from 0 to k inclusive contain the digit "0".

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Author

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Crossrefs

Programs

Python