A278038 Binary vectors not containing three consecutive 1's; or, representation of n in the tribonacci base.

0, 1, 10, 11, 100, 101, 110, 1000, 1001, 1010, 1011, 1100, 1101, 10000, 10001, 10010, 10011, 10100, 10101, 10110, 11000, 11001, 11010, 11011, 100000, 100001, 100010, 100011, 100100, 100101, 100110, 101000, 101001, 101010, 101011, 101100, 101101, 110000, 110001, 110010, 110011, 110100, 110101, 110110, 1000000

Offset

0,3

Comments

These are the nonnegative numbers written in the tribonacci numbering system.

Links

N. J. A. Sloane, Table of n, a(n) for n = 0..25280

L. Carlitz, Richard Scoville, and V. E. Hoggatt, Jr., Fibonacci Representations of Higher Order, Part 2, The Fibonacci Quarterly, Vol. 10, No. 1 (1972), pp. 43-69, 94.

F. Michel Dekking, Jeffrey Shallit, and N. J. A. Sloane, Queens in exile: non-attacking queens on infinite chess boards, Electronic J. Combin., 27:1 (2020), #P1.52.

Eric Duchêne and Michel Rigo, A morphic approach to combinatorial games: the Tribonacci case. RAIRO - Theoretical Informatics and Applications, 42, 2008, pp 375-393. See Table 2. [Also available from Numdam]

V. E. Hoggatt, Jr. and Marjorie Bicknell-Johnson, Lexicographic Ordering and Fibonacci Representations, The Fibonacci Quarterly, Vol. 20, No. 3 (1982), pp. 193-218.

Wolfdieter Lang, The Tribonacci and ABC Representations of Numbers are Equivalent, arXiv preprint arXiv:1810.09787 [math.NT], 2018.

Example

The tribonacci numbers (as in A000073(n), for n >= 3) are 1, 2, 4, 7, 13, 24, 44, 81, ... In terms of this base, 7 is written 1000, 8 is 1001, 11 is 1100, 12 is 1101, 13 is 10000, etc. Zero is 0.

Maple

# maximum index in A73 such that A73 <= n.
A73floorIdx := proc(n)
    local k ;
    for k from 3 do
        if A000073(k) = n then
            return k ;
        elif A000073(k) > n then
            return k -1 ;
        end if ;
    end do:
end proc:
A278038 := proc(n)
    local k, L, nres ;
    if n = 0 then
        0;
    else
        k := A73floorIdx(n) ;
        L := [1] ;
        nres := n-A000073(k) ;
        while k >= 4 do
            k := k-1 ;
            if nres >= A000073(k) then
                L := [1, op(L)] ;
                nres := nres-A000073(k) ;
            else
                L := [0, op(L)] ;
            end if ;
        end do:
        add( op(i, L)*10^(i-1), i=1..nops(L)) ;
    end if;
end proc:
seq(A278038(n), n=0..40) ; # R. J. Mathar, Jun 08 2022

Mathematica

t[1] = 1; t[2] = 2; t[3] = 4; t[n_] := t[n] = t[n - 1] + t[n - 2] + t[n - 3]; a[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]; FromDigits @ IntegerDigits[Total[2^(s - 1)], 2]]; Array[a, 100, 0] (* Amiram Eldar, Mar 04 2022 *)

Crossrefs

Cf. A000073, A080843 (tribonacci word, tribonacci tree).

See A003726 for the decimal representations of these binary strings.

Similar sequences: A014417 (Fibonacci), A130310 (Lucas).

Sequence in context: A066334 A136829 A262381 * A136832 A136808 A136836

Adjacent sequences: A278035 A278036 A278037 * A278039 A278040 A278041

Keyword

nonn,base

Author

N. J. A. Sloane, Nov 16 2016

Status

approved