A352103 a(n) is the maximal (or lazy) tribonacci representation of n using a binary system of vectors not containing three consecutive 0's.

0, 1, 10, 11, 100, 101, 110, 111, 1001, 1010, 1011, 1100, 1101, 1110, 1111, 10010, 10011, 10100, 10101, 10110, 10111, 11001, 11010, 11011, 11100, 11101, 11110, 11111, 100100, 100101, 100110, 100111, 101001, 101010, 101011, 101100, 101101, 101110, 101111, 110010

Offset

0,3

Comments

Each nonnegative integer has 2 unique representations as sums of distinct positive tribonacci numbers (A000073): 1, 2, 4, 7, 13, 24, ...: the minimal (or greedy, A278038) representation in which there are no 3 consecutive 1's (i.e., no 3 consecutive tribonacci numbers appear in the sum), and the maximal (or lazy) representation of n in which no 3 consecutive 0's appear.

Links

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

Formula

a(n) = A007088(A003796(n+1)).

Example

a(5) = 101 = 4 + 1.
a(6) = 110 = 4 + 2.
a(7) = 111 = 4 + 2 + 1.

Mathematica

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 == {}, 0, FromDigits[v[[i[[1, 1]] ;; -1]]]]]; Array[a, 100, 0]

Crossrefs

Cf. A000073, A007088, A003796, A278038.

Similar sequences: A104326 (Fibonacci), A130311 (Lucas).

Sequence in context: A038102 A181891 A115846 * A066334 A136829 A262381

Adjacent sequences: A352100 A352101 A352102 * A352104 A352105 A352106

Keyword

nonn,base

Author

Amiram Eldar, Mar 05 2022

Status

approved