A331467 -id:A331467 - cp's OEIS frontend

A356771 a(n) is the sum of the Fibonacci numbers in common in the Zeckendorf and dual Zeckendorf representations of n.

0, 1, 2, 0, 4, 0, 1, 7, 0, 1, 2, 3, 12, 0, 1, 2, 0, 4, 5, 6, 20, 0, 1, 2, 3, 4, 0, 1, 7, 8, 9, 10, 11, 33, 0, 1, 2, 0, 4, 5, 6, 7, 0, 1, 2, 3, 12, 13, 14, 15, 13, 17, 18, 19, 54, 0, 1, 2, 3, 4, 0, 1, 7, 8, 9, 10, 11, 12, 0, 1, 2, 0, 4, 5, 6, 20, 21, 22, 23, 24

Offset: 0

Rémy Sigrist, Aug 27 2022

The Zeckendorf and dual Zeckendorf representations both express a number n as a sum of distinct positive Fibonacci numbers; these distinct Fibonacci numbers can be encoded in binary (see A022290 for the decoding function):
- in the Zeckendorf representation (or greedy Fibonacci representation):
  - Fibonacci numbers are as big as possible (see A035517),
  - and the corresponding binary encoding, A003714(n),
    cannot have two consecutive 1's;
- in the dual Zeckendorf representation (or lazy Fibonacci representation):
  - Fibonacci numbers are as small as possible (see A112309),
  - and the corresponding binary encoding, A003754(n+1),
    cannot have two consecutive nonleading 0's.
See A356326 for a similar sequence.

			For n = 28:
- using F(k) = A000045(k),
- the Zeckendorf representation of 28 is F(8) + F(5) + F(3),
- the dual Zeckendorf representation of 28 is F(7) + F(6) + F(5) + F(3),
- F(5) and F(3) appear in both representations,
- so a(28) = F(5) + F(3) = 7.

Rémy Sigrist, Table of n, a(n) for n = 0..10946
Rémy Sigrist, PARI program
Index entries for sequences related to Zeckendorf expansion of n

Cf. A000071, A003714, A003754, A022290, A035517, A112309, A331467, A356326.

PARI
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a(n) = A022290(A003714(n) AND A003754(n+1)) (where AND denotes the bitwise AND operator).
a(n) = 0 iff n belongs to A331467.
a(n) = n iff n belongs to A000071.

A331466 The number of common terms in the Zeckendorf and dual Zeckendorf representations of n.

Original entry on oeis.org

0, 1, 1, 0, 2, 0, 1, 2, 0, 1, 1, 1, 3, 0, 1, 1, 0, 2, 1, 2, 3, 0, 1, 1, 1, 2, 0, 1, 2, 1, 2, 2, 2, 4, 0, 1, 1, 0, 2, 1, 2, 2, 0, 1, 1, 1, 3, 1, 2, 2, 1, 3, 2, 3, 4, 0, 1, 1, 1, 2, 0, 1, 2, 1, 2, 2, 2, 3, 0, 1, 1, 0, 2, 1, 2, 3, 1, 2, 2, 2, 3, 1, 2, 3, 2, 3, 3

Offset: 0

Amiram Eldar, Jan 17 2020

The indices of records are numbers of the form F(2*k - 1) - 1, for k > 0, where F(k) is the k-th Fibonacci number. The corresponding record values are k - 1 = 0, 1, 2, ...

			a(6) = 1 since the Zeckendorf representation of 6 is 1001 (i.e., F(2) + F(5)), its dual Zeckendorf representation is 111 (i.e., F(2) + F(3) + F(4)), and there is only one position with a common digit 1, corresponding to the one common summand F(2).

Amiram Eldar, Table of n, a(n) for n = 0..10000

Cf. A000045, A007895, A014417, A104326, A112310, A331467.

m = 1000; zeck = Select[Range[0, m], BitAnd[#, 2 #] == 0 &]; dualZeck = Select[Range[0, m], SequenceCount[IntegerDigits[#, 2], {0, 0}] == 0 &]; DigitCount[BitAnd[zeck[[#]], dualZeck[[#]]] & /@ Range[Min[Length[zeck], Length[dualZeck]]], 2, 1]

a(A000045(2*n - 1) - 1) = a(A000045(2*n) - 1) = n - 1.

a(A000045(n)) = a(A331467(n)) = 0 for n > 2.

A356966 Numbers with no common terms in their greedy and lazy tribonacci representations.

Original entry on oeis.org

0, 7, 13, 24, 44, 81, 88, 149, 156, 162, 274, 287, 298, 504, 511, 528, 548, 927, 934, 940, 971, 1008, 1015, 1705, 1718, 1729, 1786, 1793, 1854, 1861, 1867, 3136, 3143, 3160, 3180, 3285, 3292, 3298, 3410, 3423, 3434, 5768, 5775, 5781, 5812, 5849, 5856, 6042

Offset: 1

Rémy Sigrist, Sep 06 2022

nonn

Also numbers k such that the binary expansions of A003726(k+1) and A003796(k+1) have no common 1's.
Also positions of 0's in A356965.
This sequence is to tribonacci numbers (A000073) what A331467 is to Fibonacci numbers (A000045).
This sequence includes tribonacci numbers >= 7.

			With T = A000073:
- the greedy representation of 13 is: T(7),
- the lazy representation of 13 is: T(6) + T(5) + T(4),
- there are no common terms,
- so 13 belongs to this sequence.

Rémy Sigrist, PARI program
Index entries for sequences related to Zeckendorf expansion of n

Cf. A000045, A000073, A003726, A003796, A331467, A356965.

PARI
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A356771 a(n) is the sum of the Fibonacci numbers in common in the Zeckendorf and dual Zeckendorf representations of n.

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A331466 The number of common terms in the Zeckendorf and dual Zeckendorf representations of n.

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A356966 Numbers with no common terms in their greedy and lazy tribonacci representations.

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