A356895 -id:A356895 - cp's OEIS frontend

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

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).

A360260 a(0) = 0, and for any n > 0, let k > 0 be as small as possible and such that T(3) + ... + T(2+k) >= n (where T(m) denotes A000073(m), the m-th tribonacci number); a(n) = k + a(T(3) + ... + T(2+k) - n).

Original entry on oeis.org

0, 1, 3, 2, 5, 6, 4, 3, 8, 10, 9, 6, 7, 5, 4, 12, 11, 14, 15, 13, 8, 9, 11, 10, 7, 8, 6, 5, 16, 17, 15, 14, 19, 21, 20, 17, 18, 10, 11, 13, 12, 15, 16, 14, 9, 10, 12, 11, 8, 9, 7, 6, 21, 23, 22, 19, 20, 18, 17, 25, 24, 27, 28, 26, 21, 22, 24, 23, 12, 13, 15

Offset: 0

Rémy Sigrist, Jan 31 2023

See A356895 for the corresponding k's.

See A360259 for the Fibonacci variant.

			The first terms, alongside the corresponding k's, are:
  n   a(n)  k
  --  ----  ---
   0     0  N/A
   1     1    1
   2     3    2
   3     2    2
   4     5    3
   5     6    3
   6     4    3
   7     3    3
   8     8    4
   9    10    4
  10     9    4
  11     6    4
  12     7    4
  13     5    4
  14     4    4
  15    12    5

Rémy Sigrist, Table of n, a(n) for n = 0..10609

Cf. A000073, A027084, A356895, A360259.

tribonacci(n) = ([0,1,0; 0,0,1; 1,1,1]^n)[2,1]
{ t = k = 0; print1 (0); for (n = 1, #a = vector(70), if (n > t, t += tribonacci(2+k++);); print1 (", "a[n] = k+if (t==n, 0, a[t-n]));); }

a(A027084(n)) = n - 1.

cp's OEIS Frontend

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

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