A003726 -id:A003726 - cp's OEIS frontend

A305380 Tribonacci representation of 2^n, written in base 10.

1, 2, 4, 9, 19, 41, 88, 195, 418, 1033, 2195, 4705, 10282, 21850, 49160, 104465, 223780, 550294, 1186344, 2525345, 5514438, 11817057, 26297040, 56201282, 138856076, 295217708, 632609378, 1382640428, 2974062096, 6603081730, 14149570820, 34976354857, 74361996963

Offset: 0

N. J. A. Sloane, Jun 12 2018

Alois P. Heinz, Table of n, a(n) for n = 0..2000

Equals A003726(2^n).

Cf. A000073, A278038, A305876.

T:= proc(n) T(n):= (<<0|1|0>, <0|0|1>, <1|1|1>>^n)[2, 3] end:
b:= proc(n) option remember; local j;
      if n=0 then 0
    else for j from 2 while T(j+1)<=n do od;
         b(n-T(j))+2^(j-2)
      fi
    end:
a:= n-> b(2^n):
seq(a(n), n=0..35);  # Alois P. Heinz, Jun 12 2018

def A305380(n):
    m, tlist, s = 2**n, [1,2,4], 0
    while tlist[-1]+tlist[-2]+tlist[-3] <= m:
        tlist.append(tlist[-1]+tlist[-2]+tlist[-3])
    for d in tlist[::-1]:
        s *= 2
        if d <= m:
            s += 1
            m -= d
    return s # Chai Wah Wu, Jun 12 2018

a(9)-a(24) from Robert Israel, Jun 12 2018

Terms a(25) and beyond from Alois P. Heinz, Jun 12 2018

A356823 Tribternary numbers.

Original entry on oeis.org

0, 1, 3, 4, 9, 10, 12, 27, 28, 30, 31, 36, 37, 81, 82, 84, 85, 90, 91, 93, 108, 109, 111, 112, 243, 244, 246, 247, 252, 253, 255, 270, 271, 273, 274, 279, 280, 324, 325, 327, 328, 333, 334, 336, 729, 730, 732, 733, 738, 739, 741, 756, 757, 759, 760, 765, 766, 810, 811, 813, 814, 819

Offset: 1

Tanya Khovanova and PRIMES STEP Senior group, Aug 29 2022

These are numbers whose ternary representations consist only of zeros and ones and do not have three consecutive ones.
The sequence of Tribternary numbers can be constructed by writing out the Tribonacci representations of nonnegative integers and then evaluating the result in ternary.
These are Tribbinary numbers written in base 2 and evaluated in base 3.
These numbers are similar to Fibbinary numbers A003714, Fibternary numbers A003726, and Tribbinary numbers A060140.
Tribbinary numbers A060140 are a subsequence.
Subsequence of A005836.
The number of Tribternary numbers less than any power of three is a Tribonacci number.
We can generate this sequence recursively: start with 0 and 1; then, if x is in the sequence add 3x, 9x+1, and 27x+4 to the sequence.
The n-th Tribternary number is divisible by 3 if the n-th term of the Tribonacci word is a. Respectively, the n-th Tribbinary number is of the form 9x+1 if the n-th term of the Tribonacci word is b, and the n-th Tribbinary number is of the form 27x+4 if the n-th term of the Tribonacci word is c.
Every nonnegative integer can be written as a sum of three Tribternary numbers.
Every number has a Tribternary multiple.

Cf. A003714, A003726, A005836, A060140.

Select[Range[0, 1000], SequenceCount[IntegerDigits[#, 3], {1, 1, 1}] == 0 && SequenceCount[IntegerDigits[#, 3], {2}] == 0 &]

import heapq
from itertools import islice
def agen(): # generator of terms, using recursion in Comments
    x, h = None, [0]
    while True:
        x, oldx = heapq.heappop(h), x
        if x != oldx:
            yield x
            for t in [3*x, 9*x+1, 27*x+4]: heapq.heappush(h, t)
print(list(islice(agen(), 62))) # Michael S. Branicky, Aug 30 2022

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
```
See Links section.
```

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
```
See Links section.
```

A305378 Tribonacci representation of 2n+1, written in base 10.

Original entry on oeis.org

1, 3, 5, 8, 10, 12, 16, 18, 20, 22, 25, 27, 33, 35, 37, 40, 42, 44, 48, 50, 52, 54, 65, 67, 69, 72, 74, 76, 80, 82, 84, 86, 89, 91, 97, 99, 101, 104, 106, 108, 128, 130, 132, 134, 137, 139, 141, 145, 147, 149, 152, 154, 160, 162, 164, 166, 169, 171, 173, 177, 179, 181, 192, 194, 196, 198, 201

Offset: 0

N. J. A. Sloane, Jun 12 2018

Robert Israel, Table of n, a(n) for n = 0..10000

Equals A003726(2n+1).

Cf. A278038.

L[0]:= [0]: L[1]:= [1]:
for d from 2 to 10 do
  L[d]:= map(t -> (2*t, `if`(t mod 4 <> 3, 2*t+1,NULL)), L[d-1])
od:
A003726:=map(op,[seq(L[i],i=0..10)]):
seq(A003726[i],i=2..nops(A003726),2); # Robert Israel, Jun 12 2018

def A305378(n):
    m, tlist, s = 2*n+1, [1,2,4], 0
    while tlist[-1]+tlist[-2]+tlist[-3] <= m:
        tlist.append(tlist[-1]+tlist[-2]+tlist[-3])
    for d in tlist[::-1]:
        s *= 2
        if d <= m:
            s += 1
            m -= d
    return s # Chai Wah Wu, Jun 12 2018

More terms from Robert Israel, Jun 12 2018

A305878 For any number n >= 0: apply the map 0 -> "0", 1 -> "01", 2 -> "011" to the ternary representation of n and interpret the result as a binary string.

Original entry on oeis.org

0, 1, 3, 2, 5, 11, 6, 13, 27, 4, 9, 19, 10, 21, 43, 22, 45, 91, 12, 25, 51, 26, 53, 107, 54, 109, 219, 8, 17, 35, 18, 37, 75, 38, 77, 155, 20, 41, 83, 42, 85, 171, 86, 173, 347, 44, 89, 179, 90, 181, 363, 182, 365, 731, 24, 49, 99, 50, 101, 203, 102, 205, 411

Offset: 0

Rémy Sigrist, Jun 13 2018

This sequence is a ternary analog of A048678.

This sequence is a permutation of A003726.

			The first terms, alongside the ternary representation of n and the binary representation of a(n), are:
  n   a(n)  tern(n)  bin(a(n))
  --  ----  -------  ---------
   0     0        0          0
   1     1        1          1
   2     3        2         11
   3     2       10         10
   4     5       11        101
   5    11       12       1011
   6     6       20        110
   7    13       21       1101
   8    27       22      11011
   9     4      100        100
  10     9      101       1001
  11    19      102      10011
  12    10      110       1010

Rémy Sigrist, Colored logarithmic scatterplot of the first 3^10 terms (where the color is function of A053735(n) + A081604(n))

Cf. A000120, A003726, A048678, A053735, A081604.

a(n) = if (n==0, 0, my (d=n%3); a(n\3) * 2^(d+1) + (2^d-1))

a(0) = 0.
a(3*n) = 2*a(n).
a(3*n + 1) = 4*a(n) + 1.
a(3*n + 2) = 8*a(n) + 3.
A000120(a(n)) = A053735(n).

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
```
See Links section.
```

T(n, 1) = A003796(n+1).

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

cp's OEIS Frontend

A305380 Tribonacci representation of 2^n, written in base 10.

Views

Author

Keywords

Links

Crossrefs

Programs

Maple

Python

Extensions

A356823 Tribternary numbers.

Views

Author

Keywords

Comments

Crossrefs

Programs

Mathematica

Python

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

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

PARI

Formula

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

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

PARI

A305378 Tribonacci representation of 2n+1, written in base 10.

Views

Author

Keywords

Links

Crossrefs

Programs

Maple

Python

Extensions

A305878 For any number n >= 0: apply the map 0 -> "0", 1 -> "01", 2 -> "011" to the ternary representation of n and interpret the result as a binary string.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

PARI

Formula

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.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

PARI

Formula