A361755 -id:A361755 - cp's OEIS frontend

A361756 Irregular triangle T(n, k), n >= 0, k = 1..A361757(n), read by rows; the n-th row lists the numbers k such that the Fibonacci numbers that appear in the dual Zeckendorf representation of k also appear in that of n.

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

Offset: 0

Rémy Sigrist, Mar 23 2023

In other words, the n-th row lists the numbers k such that A003754(1+n) AND A003754(1+k) = A003754(1+k) (where AND denotes the bitwise AND operator).

The dual Zeckendorf representation is also known as the lazy 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, 1, 2, 3
   4  0, 1, 4
   5  0, 2, 5
   6  0, 1, 2, 3, 4, 5, 6
   7  0, 2, 7
   8  0, 1, 2, 3, 7, 8
   9  0, 1, 4, 9
  10  0, 2, 5, 7, 10
  11  0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11
  12  0, 1, 4, 12

Rémy Sigrist, Table of n, a(n) for n = 0..9956 (rows for n = 0..377 flattened)
Rémy Sigrist, PARI program
Index entries for sequences related to Zeckendorf expansion of n

See A361755 for a similar sequence.

Cf. A003754, A003842, A356771, A361757.

PARI
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See Links section.
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T(n, 1) = 0.
T(n, 2) = A003842(n - 1) for any n > 0.
T(n, A361757(n)) = n.

A362755 Irregular triangle read by rows; the n-th row lists the numbers k such that if phi^e appears in the base phi expansion of k then phi^e also appears in the base phi expansion of n (where phi denotes A001622, the golden ratio).

Original entry on oeis.org

0, 0, 1, 0, 2, 0, 3, 0, 1, 3, 4, 0, 5, 0, 6, 0, 7, 0, 1, 7, 8, 0, 2, 7, 9, 0, 3, 7, 10, 0, 1, 3, 4, 7, 8, 10, 11, 0, 12, 0, 13, 0, 14, 0, 1, 14, 15, 0, 16, 0, 17, 0, 18, 0, 1, 18, 19, 0, 2, 18, 20, 0, 3, 18, 21, 0, 1, 3, 4, 18, 19, 21, 22, 0, 5, 18, 23, 0, 6, 18, 24

Offset: 0

Rémy Sigrist, May 02 2023

See A361755 for a similar sequence.

			Triangle 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, 6
   7  0, 7
   8  0, 1, 7, 8
   9  0, 2, 7, 9
  10  0, 3, 7, 10
  11  0, 1, 3, 4, 7, 8, 10, 11
  12  0, 12
  13  0, 13
  14  0, 14
  15  0, 1, 14, 15

Rémy Sigrist, Table of n, a(n) for n = 0..9999 (rows for n = 0..1060 flattened)
Rémy Sigrist, PARI program
Wikipedia, Golden ratio base

Cf. A001622, A104605, A214971, A361755.

PARI
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See Links section.
```

T(n, 1) = 0.

T(n, 2) = 1 iff n belongs to A214971.

cp's OEIS Frontend

A361756 Irregular triangle T(n, k), n >= 0, k = 1..A361757(n), read by rows; the n-th row lists the numbers k such that the Fibonacci numbers that appear in the dual Zeckendorf representation of k also appear in that of n.

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