A356898 -id:A356898 - cp's OEIS frontend

A356896 Nonnegative numbers whose maximal tribonacci representation (A352103) ends in an even number of 1's.

0, 2, 3, 4, 6, 9, 10, 11, 13, 14, 15, 16, 17, 19, 22, 23, 24, 26, 28, 30, 33, 34, 35, 37, 38, 39, 40, 41, 43, 46, 47, 48, 50, 51, 53, 54, 55, 57, 58, 59, 60, 61, 63, 66, 67, 68, 70, 72, 74, 77, 78, 79, 81, 82, 83, 84, 85, 87, 90, 91, 92, 94, 96, 97, 98, 100, 103

Offset: 1

Amiram Eldar, Sep 03 2022

Numbers k such that A356898(k) is even.

The asymptotic density of this sequence is c/(c+1) = 0.647798..., where c = 1.839286... (A058265) is the tribonacci constant.

			   n  a(n)  A352103(n)  A356898(n)
   -  ----  ----------  ----------
   1     0           0          0
   2     2          10          0
   3     3          11          2
   4     4         100          0
   5     6         110          0
   6     9        1010          0
   7    10        1011          2
   8    11        1100          0
   9    13        1110          0
  10    14        1111          4

Amiram Eldar, Table of n, a(n) for n = 1..10000

Complement of A356897.
Cf. A058265, A352103, A356898.
Similar sequences: A308197, A342051.

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]]; f[v_] := Module[{m = Length[v], k}, k = m; While[v[[k]] == 1, k--]; m - k]; c[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 == {}, 0, f[v[[i[[1, 1]] ;; -1]]], 10]]; Select[Range[0, 100], EvenQ[c[#]] &]

A356897 Nonnegative numbers whose maximal tribonacci representation (A352103) ends in an odd number of 1's.

Original entry on oeis.org

1, 5, 7, 8, 12, 18, 20, 21, 25, 27, 29, 31, 32, 36, 42, 44, 45, 49, 52, 56, 62, 64, 65, 69, 71, 73, 75, 76, 80, 86, 88, 89, 93, 95, 99, 101, 102, 106, 108, 110, 112, 113, 117, 123, 125, 126, 130, 133, 137, 143, 145, 146, 150, 152, 154, 156, 157, 161, 167, 169

Offset: 1

Amiram Eldar, Sep 03 2022

Numbers k such that A356898(k) is odd.

The asymptotic density of this sequence is 1/(c+1) = 0.352201..., where c = 1.839286... (A058265) is the tribonacci constant.

			   n  a(n)  A352103(n)  A356898(n)
   -  ----  ----------  ----------
   1     1           1          1
   2     5         101          1
   3     7         111          3
   4     8        1001          1
   5    12        1101          1
   6    18       10101          1
   7    20       10111          3
   8    21       11001          1
   9    25       11101          1
  10    27       11111          5

Amiram Eldar, Table of n, a(n) for n = 1..10000

Complement of A356896.
Cf. A058265, A352103, A356898.
Similar sequences: A001950, A308198, A342050.

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]]; f[v_] := Module[{m = Length[v], k}, k = m; While[v[[k]] == 1, k--]; m - k]; c[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 == {}, 0, f[v[[i[[1, 1]] ;; -1]]], 10]]; Select[Range[0, 200], OddQ[c[#]] &]

cp's OEIS Frontend

A356896 Nonnegative numbers whose maximal tribonacci representation (A352103) ends in an even number of 1's.

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