A278042 -id:A278042 - cp's OEIS frontend

A278043 Number of 1's in tribonacci representation of n (cf. A278038).

0, 1, 1, 2, 1, 2, 2, 1, 2, 2, 3, 2, 3, 1, 2, 2, 3, 2, 3, 3, 2, 3, 3, 4, 1, 2, 2, 3, 2, 3, 3, 2, 3, 3, 4, 3, 4, 2, 3, 3, 4, 3, 4, 4, 1, 2, 2, 3, 2, 3, 3, 2, 3, 3, 4, 3, 4, 2, 3, 3, 4, 3, 4, 4, 3, 4, 4, 5, 2, 3, 3, 4, 3, 4, 4, 3, 4, 4, 5, 4, 5, 1, 2, 2, 3, 2, 3, 3, 2, 3, 3, 4, 3, 4, 2, 3, 3, 4, 3, 4

Offset: 0

N. J. A. Sloane, Nov 18 2016

N. J. A. Sloane, Table of n, a(n) for n = 0..20000

Cf. A001590, A003726, A007895, A278038, A278042, A278044.

t[1] = 1; t[2] = 2; t[3] = 4; t[n_] := t[n] = t[n - 1] + t[n - 2] + t[n - 3]; a[0] = 0; a[n_] := Module[{s = {}, m = n, k}, While[m > 0, k = 1; While[t[k] <= m, k++]; k--; AppendTo[s, k]; m -= t[k]; k = 1]; DigitCount[Total[2^(s - 1)], 2, 1]]; Array[a, 100, 0] (* Amiram Eldar, Mar 04 2022 *)

a(n) = A000120(A003726(n+1)). - John Keith, May 23 2022

A278045 Number of trailing 0's in tribonacci representation of n (cf. A278038).

Original entry on oeis.org

1, 0, 1, 0, 2, 0, 1, 3, 0, 1, 0, 2, 0, 4, 0, 1, 0, 2, 0, 1, 3, 0, 1, 0, 5, 0, 1, 0, 2, 0, 1, 3, 0, 1, 0, 2, 0, 4, 0, 1, 0, 2, 0, 1, 6, 0, 1, 0, 2, 0, 1, 3, 0, 1, 0, 2, 0, 4, 0, 1, 0, 2, 0, 1, 3, 0, 1, 0, 5, 0, 1, 0, 2, 0, 1, 3, 0, 1, 0, 2, 0, 7, 0, 1, 0, 2, 0, 1, 3, 0, 1, 0, 2, 0, 4, 0, 1, 0, 2, 0

Offset: 0

N. J. A. Sloane, Nov 18 2016

The number mod 3 of trailing 0's in the tribonacci representation of n  >= 1 (this sequence mod 3) is the tribonacci word itself (A080843). - N. J. A. Sloane, Oct 04 2018
The number of trailing 1's in the tribonacci representation of n >= 0 (cf. A278038) is also the tribonacci word itself (A080843).
From Amiram Eldar, Mar 04 2022: (Start)
The asymptotic density of the occurrences of k = 0, 1, 2, ... is (c-1)/c^(k+1), where c = 1.839286... (A058265) is the tribonacci constant.
The asymptotic mean of this sequence is 1/(c-1) = 1.191487... (End)

N. J. A. Sloane, Table of n, a(n) for n = 0..20000

Cf. A278038, A278042, A278043, A278044, A080843, A058265.

t[1] = 1; t[2] = 2; t[3] = 4; t[n_] := t[n] = t[n - 1] + t[n - 2] + t[n - 3]; a[0] = 1; a[n_] := Module[{s = {}, m = n, k}, While[m > 0, k = 1; While[t[k] <= m, k++]; k--; AppendTo[s, k]; m -= t[k]; k = 1]; Min[s] - 1]; Array[a, 100, 0] (* Amiram Eldar, Mar 04 2022 *)

A278044 Length of tribonacci representation of n (cf. A278038).

Original entry on oeis.org

1, 1, 2, 2, 3, 3, 3, 4, 4, 4, 4, 4, 4, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8

Offset: 0

N. J. A. Sloane, Nov 18 2016

For n>=2, n appears A001590(n+2) times. - John Keith, May 23 2022

N. J. A. Sloane, Table of n, a(n) for n = 0..20000
Wolfdieter Lang, The Tribonacci and ABC Representations of Numbers are Equivalent, arXiv preprint arXiv:1810.09787 [math.NT], 2018.

Cf. A278038, A278042, A278043.
Cf. A001590.
Similar to, but strictly different from, A201052.

t[1] = 1; t[2] = 2; t[3] = 4; t[n_] := t[n] = t[n - 1] + t[n - 2] + t[n - 3]; a[0] = 1; a[n_] := Module[{k = 1}, While[t[k] <= n, k++]; k - 1]; Array[a, 100, 0] (* Amiram Eldar, Mar 04 2022 *)

a(n) = A278042(n) + A278043(n).

A356894 a(n) is the number of 0's in the maximal tribonacci representation of n (A352103).

Original entry on oeis.org

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

Offset: 0

Amiram Eldar, Sep 03 2022

			  n  a(n)  A352103(n)
  -  ----  ----------
  0     1           0
  1     0           1
  2     1          10
  3     0          11
  4     2         100
  5     1         101
  6     1         110
  7     0         111
  8     2        1001
  9     2        1010

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

Cf. A000073, A352103, A352104, A356895.

Similar sequences: A023416, A102364, A117479, A278042.

t[1] = 1; t[2] = 2; t[3] = 4; t[n_] := t[n] = t[n - 1] + t[n - 2] + t[n - 3]; trib[n_] := Module[{s = {}, m = n, k}, While[m > 0, k = 1; While[t[k] <= m, k++]; k--; AppendTo[s, k]; m -= t[k]; k = 1]; IntegerDigits[Total[2^(s - 1)], 2]]; a[n_] := Module[{v = trib[n]}, nv = Length[v]; i = 1; While[i <= nv - 3, If[v[[i ;; i + 3]] == {1, 0, 0, 0}, v[[i ;; i + 3]] = {0, 1, 1, 1}; If[i > 3, i -= 4]]; i++]; i = Position[v, _?(# > 0 &)]; If[i == {}, 1, Count[v[[i[[1, 1]] ;; -1]], 0]]]; Array[a, 100, 0]

a(n) = A356895(n) - A352104(n).

cp's OEIS Frontend

A278043 Number of 1's in tribonacci representation of n (cf. A278038).

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A278045 Number of trailing 0's in tribonacci representation of n (cf. A278038).

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A278044 Length of tribonacci representation of n (cf. A278038).

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A356894 a(n) is the number of 0's in the maximal tribonacci representation of n (A352103).

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