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A352103
a(n) is the maximal (or lazy) tribonacci representation of n using a binary system of vectors not containing three consecutive 0's.
17
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
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
Similar sequences: A104326 (Fibonacci), A130311 (Lucas).
Sequence in context: A038102 A181891 A115846 * A066334 A136829 A262381
KEYWORD
nonn,base
AUTHOR
Amiram Eldar, Mar 05 2022
STATUS
approved