A381607 -id:A381607 - cp's OEIS frontend

A381609 a(n) is the number of occurrences of n in A381607.

1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 2, 2, 2, 2, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 2, 2, 2, 2, 1, 1, 1, 1, 2, 1, 1, 1, 1, 2, 2, 2, 3, 2, 2, 2, 2, 3, 2, 2, 2, 1, 1, 1, 1, 2, 1, 1, 1, 1, 2, 2, 2, 2, 1, 1, 1, 1, 2, 1, 1

Offset: 0

Rémy Sigrist, Mar 01 2025

All terms are positive (see A381579).

			The number 8 appears twice in A381607, so a(8) = 2.

Rémy Sigrist, Table of n, a(n) for n = 0..10000
Rémy Sigrist, PARI program

Cf. A381579, A381607.

PARI
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a(n) > 0.

A381610 Irregular table T(n, k), n >= 0, k = 1..A381609(n), read by rows: the n-th row lists the numbers m such that A381607(m) = n.

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 27, 24, 28, 25, 29, 26, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 54, 51, 55, 52, 56, 53, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67

Offset: 0

Rémy Sigrist, Mar 01 2025

As a flat sequence, this is a permutation of the nonnegative integers with inverse A381611.

			Table T(n, k) begins:
  n   n-th row
  --  --------
   0  0
   1  1
   2  2
   3  3
   4  4
   5  5
   6  6
   7  7
   8  8, 9
   9  10
  10  11
  11  12
  12  13
  13  14
  14  15
  15  16
  16  17, 18
  17  19
  18  20
  19  21
  20  22

Rémy Sigrist, Table of n, a(n) for n = 0..4693 (rows for n = 0..3^7 flattened)
Rémy Sigrist, PARI program
Index entries for sequences that are permutations of the natural numbers

Cf. A381607, A381609, A381611 (inverse).

PARI
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A381618 Reverse the Chung-Graham representation of n while preserving its trailing zeros: a(n) = A381607(A263273(A381608(n))).

Original entry on oeis.org

0, 1, 2, 3, 4, 7, 6, 5, 8, 9, 17, 11, 12, 20, 19, 15, 16, 10, 18, 14, 13, 21, 22, 43, 24, 30, 51, 45, 38, 29, 25, 46, 32, 33, 54, 53, 41, 50, 28, 49, 40, 36, 42, 23, 44, 27, 31, 52, 48, 39, 37, 26, 47, 35, 34, 55, 56, 111, 58, 77, 132, 113, 98, 63, 64, 119, 79

Offset: 0

Rémy Sigrist, Mar 02 2025

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

			The first terms, alongside their Chung-Graham representation, are:
  n   a(n)  A381579(n)  A381579(a(n))
  --  ----  ----------  -------------
   0     0           0              0
   1     1           1              1
   2     2           2              2
   3     3          10             10
   4     4          11             11
   5     7          12             21
   6     6          20             20
   7     5          21             12
   8     8         100            100
   9     9         101            101
  10    17         102            201
  11    11         110            110
  12    12         111            111
  13    20         112            211
  14    19         120            210
  15    15         121            121
  16    16         200            200

See A345201 for a similar sequence.

Cf. A000045, A263273, A381579, A381607, A381608.

A381607(n) = { my (t = Vecrev(digits(n, 3))); sum(k = 1, #t, t[k] * fibonacci(2*k)); }
A263273(n) = { my (t = 3^if (n, valuation(n, 3), 0)); t * fromdigits(Vecrev(digits(n / t, 3)), 3) }
A381608(n) = { for (k = 1, oo, my (f = fibonacci(2*k)); if (f >= n, my (v = 0); while (n, while (n >= f, n -= f; v += 3^(k-1);); f = fibonacci(2*k--);); return (v););); }
a(n) = A381607(A263273(A381608(n)))

a(n) <= A000045(2*k) iff n <= A000045(2*k).

A381608 Nonnegative integers whose ternary expansion does not contain pairs of 2's only separated by (zero or more) 1's, with offset 0.

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 6, 7, 9, 10, 11, 12, 13, 14, 15, 16, 18, 19, 20, 21, 22, 27, 28, 29, 30, 31, 32, 33, 34, 36, 37, 38, 39, 40, 41, 42, 43, 45, 46, 47, 48, 49, 54, 55, 56, 57, 58, 59, 60, 61, 63, 64, 65, 66, 67, 81, 82, 83, 84, 85, 86, 87, 88, 90, 91, 92, 93

Offset: 0

Rémy Sigrist, Mar 01 2025

The ternary expansion of a(n) equals the decimal expansion of A381579(n).

			The ternary expansion of 20, "211", has no pairs of 2's, so 20 belongs to the sequence.The ternary expansion of 21, "212", has a pair of 2s only separated by 1's, so 21 does not belong to the sequence.

Rémy Sigrist, Table of n, a(n) for n = 0..6560

Cf. A028898, A381579, A381607.

a(n) = { for (k = 1, oo, my (f = fibonacci(2*k)); if (f >= n, my (v = 0); while (n, while (n >= f, n -= f; v += 3^(k-1);); f = fibonacci(2*k--);); return (v););); }

a(n) = A028898(A381579(n)).

A381607(a(n)) = n.

cp's OEIS Frontend

A381609 a(n) is the number of occurrences of n in A381607.

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A381610 Irregular table T(n, k), n >= 0, k = 1..A381609(n), read by rows: the n-th row lists the numbers m such that A381607(m) = n.

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A381618 Reverse the Chung-Graham representation of n while preserving its trailing zeros: a(n) = A381607(A263273(A381608(n))).

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A381608 Nonnegative integers whose ternary expansion does not contain pairs of 2's only separated by (zero or more) 1's, with offset 0.

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