A361756 -id:A361756 - cp's OEIS frontend

A361757 a(n) is the number of terms in the n-th row of A361756.

1, 2, 2, 4, 3, 3, 7, 3, 6, 4, 5, 12, 4, 4, 10, 4, 8, 6, 8, 20, 4, 8, 5, 7, 17, 5, 5, 13, 6, 12, 9, 13, 33, 5, 5, 13, 5, 10, 8, 11, 28, 5, 10, 6, 9, 22, 7, 7, 19, 9, 18, 14, 21, 54, 5, 10, 6, 9, 22, 6, 6, 16, 8, 16, 12, 18, 46, 6, 6, 16, 6, 12, 10, 14, 36, 7

Offset: 0

Rémy Sigrist, Mar 23 2023

Rémy Sigrist, Table of n, a(n) for n = 0..10945
Rémy Sigrist, PARI program
Index entries for sequences related to Zeckendorf expansion of n

Cf. A001911, A112310, A361756.

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a(n) <= n + 1 with equality iff n belongs to A001911.

a(n) >= A112310(n) + 1.

A361755 Irregular triangle T(n, k), n >= 0, k = 1..2^A007895(n), read by rows; the n-th row lists the numbers k such that the Fibonacci numbers that appear in the Zeckendorf representation of k also appear in that of n.

Original entry on oeis.org

0, 0, 1, 0, 2, 0, 3, 0, 1, 3, 4, 0, 5, 0, 1, 5, 6, 0, 2, 5, 7, 0, 8, 0, 1, 8, 9, 0, 2, 8, 10, 0, 3, 8, 11, 0, 1, 3, 4, 8, 9, 11, 12, 0, 13, 0, 1, 13, 14, 0, 2, 13, 15, 0, 3, 13, 16, 0, 1, 3, 4, 13, 14, 16, 17, 0, 5, 13, 18, 0, 1, 5, 6, 13, 14, 18, 19, 0, 2, 5, 7, 13, 15, 18, 20

Offset: 0

Rémy Sigrist, Mar 23 2023

In other words, the n-th row lists the numbers k such that A003714(n) AND A003714(k) = A003714(k) (where AND denotes the bitwise AND operator).

The Zeckendorf representation is also known as the greedy Fibonacci representation (see A356771 for further details).

			Triangle T(n, k) begins:
  n   n-th row
  --  ------------------------
   0  0
   1  0, 1
   2  0, 2
   3  0, 3
   4  0, 1, 3, 4
   5  0, 5
   6  0, 1, 5, 6
   7  0, 2, 5, 7
   8  0, 8
   9  0, 1, 8, 9
  10  0, 2, 8, 10
  11  0, 3, 8, 11
  12  0, 1, 3, 4, 8, 9, 11, 12

Rémy Sigrist, Table of n, a(n) for n = 0..10924 (rows for n = 0..610 flattened)
Rémy Sigrist, PARI program
Index entries for sequences related to Zeckendorf expansion of n

See A361756 for a similar sequence.

Cf. A003714, A007895, A139764, A295989, A356771.

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T(n, 1) = 0.
T(n, 2) = A139764(n) for any n > 0.
T(n, 2^A007895(n)) = n.

A361789 A(n, k) is the sum of the distinct terms in the dual Zeckendorf representations of n or of k; square array A(n, k) read by antidiagonals, n, k >= 0.

Original entry on oeis.org

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

Offset: 0

Rémy Sigrist, Mar 24 2023

The dual Zeckendorf representation corresponds to the lazy Fibonacci representation (see A356771 for further details).

			Array A(n, k) begins:
  n\k |  0   1   2   3   4   5   6   7   8   9  10  11  12  13
  ----+-------------------------------------------------------
    0 |  0   1   2   3   4   5   6   7   8   9  10  11  12  13
    1 |  1   1   3   3   4   6   6   8   8   9  11  11  12  14
    2 |  2   3   2   3   6   5   6   7   8  11  10  11  14  13
    3 |  3   3   3   3   6   6   6   8   8  11  11  11  14  14
    4 |  4   4   6   6   4   6   6  11  11   9  11  11  12  14
    5 |  5   6   5   6   6   5   6  10  11  11  10  11  14  13
    6 |  6   6   6   6   6   6   6  11  11  11  11  11  14  14
    7 |  7   8   7   8  11  10  11   7   8  11  10  11  19  18
    8 |  8   8   8   8  11  11  11   8   8  11  11  11  19  19
    9 |  9   9  11  11   9  11  11  11  11   9  11  11  17  19
   10 | 10  11  10  11  11  10  11  10  11  11  10  11  19  18
   11 | 11  11  11  11  11  11  11  11  11  11  11  11  19  19
   12 | 12  12  14  14  12  14  14  19  19  17  19  19  12  14
   13 | 13  14  13  14  14  13  14  18  19  19  18  19  14  13

Rémy Sigrist, Colored representation of the table for x, y < Fibonacci(16) - 1 (where the color is function of A(x, y))
Rémy Sigrist, PARI program
Index entries for sequences related to Zeckendorf expansion of n

Cf. A003754, A003986, A022290, A334348, A356771, A356969, A361756.

PARI
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A(n, k) = A022290(A003754(n+1) OR A003754(k+1)) (where OR denotes the bitwise OR operator, A004198).
A(n, k) = A(k, n).
A(n, 0) = n.
A(n, n) = n.
A(A(m, n), k) = A(m, A(n, k)).
A(A(n, k), n) = A(n, k).
A(n, A361756(n, k)) = n.

cp's OEIS Frontend

A361757 a(n) is the number of terms in the n-th row of A361756.

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A361755 Irregular triangle T(n, k), n >= 0, k = 1..2^A007895(n), read by rows; the n-th row lists the numbers k such that the Fibonacci numbers that appear in the Zeckendorf representation of k also appear in that of n.

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A361789 A(n, k) is the sum of the distinct terms in the dual Zeckendorf representations of n or of k; square array A(n, k) read by antidiagonals, n, k >= 0.

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Comments

Examples

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Crossrefs

Programs

PARI

Formula