A003796 -id:A003796 - cp's OEIS frontend

A356964 Replace 2^k in binary expansion of n with tribonacci(k+3) (where tribonacci corresponds to A000073).

0, 1, 2, 3, 4, 5, 6, 7, 7, 8, 9, 10, 11, 12, 13, 14, 13, 14, 15, 16, 17, 18, 19, 20, 20, 21, 22, 23, 24, 25, 26, 27, 24, 25, 26, 27, 28, 29, 30, 31, 31, 32, 33, 34, 35, 36, 37, 38, 37, 38, 39, 40, 41, 42, 43, 44, 44, 45, 46, 47, 48, 49, 50, 51, 44, 45, 46, 47

Offset: 0

Rémy Sigrist, Sep 06 2022

This sequence is to tribonacci numbers (A000073) what A022290 is to Fibonacci numbers (A000045).

For any k >= 0, k appears A117546(k) times in this sequence.

			For n = 9:
- 9 = 2^3 + 2^0,
- so a(9) = A000073(3+3) + A000073(0+3) = 7 + 1 = 8.

Rémy Sigrist, Table of n, a(n) for n = 0..8192
Index entries for sequences related to Zeckendorf expansion of n

Cf. A000073, A003726, A003796, A022290, A117546.

a(n) = { my (v=0, k); while (n, n-=2^k=valuation(n,2); v+=([0,1,0; 0,0,1; 1,1,1]^(3+k))[2,1]); return (v); }

def A356964(n):
    a, b, c, s = 1,2,4,0
    for i in bin(n)[-1:1:-1]:
        s += int(i)*a
        a, b, c = b, c, a+b+c
    return s # Chai Wah Wu, Sep 10 2022

a(A003726(n+1)) = n.

a(A003796(n+1)) = n.

A356965 a(n) is the sum of the tribonacci numbers in common in the greedy and lazy tribonacci representations of n.

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 6, 0, 8, 9, 10, 11, 12, 0, 1, 15, 16, 17, 18, 19, 13, 21, 22, 23, 0, 1, 2, 3, 28, 29, 30, 24, 32, 33, 34, 35, 36, 24, 25, 39, 40, 41, 42, 43, 0, 1, 2, 3, 4, 5, 6, 7, 52, 53, 54, 55, 56, 44, 45, 59, 60, 61, 62, 63, 57, 65, 66, 67, 44, 45, 46

Offset: 0

Rémy Sigrist, Sep 06 2022

This sequence is to tribonacci numbers (A000073) what A356771 is to Fibonacci numbers (A000045).

			For n = 58:
- with T = A000073,
- the greedy representation of 58 is: T(9) + T(7) + T(3),
- the lazy representation of 58 is: T(9) + T(6) + T(5) + T(4) + T(3),
- so a(59) = T(9) + T(3) = 45.

Rémy Sigrist, Table of n, a(n) for n = 0..10609
Rémy Sigrist, PARI program
Index entries for sequences related to Zeckendorf expansion of n

Cf. A000045, A000073, A003726, A003796, A356771, A356899, A356964, A356966.

PARI
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See Links section.
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a(n) = A356964(A003796(n+1) AND A003726(n+1)) (where AND denotes the bitwise AND operator).
a(n) <= n with equality iff n belongs to A356899.
a(n) = 0 iff n belongs to A356966.

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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See Links section.
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A356974 Irregular triangle T(n, k) read by rows, n >= 0, k = 1..A117546(n); the n-th row contains the numbers m such that A356964(m) = n, in increasing order.

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 16, 15, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 32, 29, 33, 30, 34, 31, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 48, 47, 49, 50, 51, 52, 53, 54, 55, 56, 64, 57, 65, 58, 66, 59, 67, 60, 68, 61, 69, 62, 70

Offset: 0

Rémy Sigrist, Sep 07 2022

This sequence is to tribonacci numbers (A000073) what A345101 is to Fibonacci numbers (A000045).

This sequence (when interpreted as a flat sequence) is a permutation of the nonnegative integers.

			Triangle begins:
     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, 16]
    14   [15, 17]

Rémy Sigrist, Table of n, a(n) for n = 0..7021
Rémy Sigrist, PARI program
Index entries for sequences related to Zeckendorf expansion of n
Index entries for sequences that are permutations of the natural numbers

Cf. A000045, A000073, A003726, A003796, A117546 (row lengths), A345101, A356964.

PARI
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See Links section.
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T(n, 1) = A003796(n+1).

T(n, A117546(n)) = A003726(n+1).

cp's OEIS Frontend

A356964 Replace 2^k in binary expansion of n with tribonacci(k+3) (where tribonacci corresponds to A000073).

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A356965 a(n) is the sum of the tribonacci numbers in common in the greedy and lazy tribonacci representations of n.

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

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A356974 Irregular triangle T(n, k) read by rows, n >= 0, k = 1..A117546(n); the n-th row contains the numbers m such that A356964(m) = n, in increasing order.

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