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
See A003726 for the decimal representations of these binary strings.
KEYWORD
nonn,base
AUTHOR
N. J. A. Sloane, Nov 16 2016
STATUS
approved