A308198 -id:A308198 - cp's OEIS frontend

A308197 Numbers m such that the tribonacci representation of m (A278038) ends in an even number of 0's.

1, 3, 4, 5, 8, 10, 11, 12, 13, 14, 16, 17, 18, 21, 23, 25, 27, 28, 29, 32, 34, 35, 36, 37, 38, 40, 41, 42, 44, 45, 47, 48, 49, 52, 54, 55, 56, 57, 58, 60, 61, 62, 65, 67, 69, 71, 72, 73, 76, 78, 79, 80, 82, 84, 85, 86, 89, 91, 92, 93, 94, 95, 97, 98, 99, 102, 104, 106, 108, 109, 110, 113, 115, 116, 117, 118, 119, 121

Offset: 1

N. J. A. Sloane, Jun 22 2019

The asymptotic density of this sequence is c/(c+1) = 0.647798..., where c = 1.839286... (A058265) is the tribonacci constant. - Amiram Eldar, Mar 04 2022

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

Cf. A278038, A278045, A308198, A058265.

t[1] = 1; t[2] = 2; t[3] = 4; t[n_] := t[n] = t[n - 1] + t[n - 2] + t[n - 3]; q[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]; EvenQ[Min[s] - 1]]; Select[Range[0, 121], q] (* Amiram Eldar, Mar 04 2022 *)

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

A308197 Numbers m such that the tribonacci representation of m (A278038) ends in an even number of 0's.

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