This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A278038 #42 Jan 05 2025 19:51:41
%S A278038 0,1,10,11,100,101,110,1000,1001,1010,1011,1100,1101,10000,10001,
%T A278038 10010,10011,10100,10101,10110,11000,11001,11010,11011,100000,100001,
%U A278038 100010,100011,100100,100101,100110,101000,101001,101010,101011,101100,101101,110000,110001,110010,110011,110100,110101,110110,1000000
%N A278038 Binary vectors not containing three consecutive 1's; or, representation of n in the tribonacci base.
%C A278038 These are the nonnegative numbers written in the tribonacci numbering system.
%H A278038 N. J. A. Sloane, <a href="/A278038/b278038.txt">Table of n, a(n) for n = 0..25280</a>
%H A278038 L. Carlitz, Richard Scoville, and V. E. Hoggatt, Jr., <a href="https://web.archive.org/web/2024*/https://www.fq.math.ca/Scanned/10-1/carlitz3-a.pdf">Fibonacci Representations of Higher Order</a>, <a href="https://web.archive.org/web/2024*/https://www.fq.math.ca/Scanned/10-1/carlitz3-b.pdf">Part 2</a>, The Fibonacci Quarterly, Vol. 10, No. 1 (1972), pp. 43-69, 94.
%H A278038 F. Michel Dekking, Jeffrey Shallit, and N. J. A. Sloane, <a href="https://www.combinatorics.org/ojs/index.php/eljc/article/view/v27i1p52/8039">Queens in exile: non-attacking queens on infinite chess boards</a>, Electronic J. Combin., 27:1 (2020), #P1.52.
%H A278038 Eric Duchêne and Michel Rigo, <a href="http://dx.doi.org/10.1051/ita:2007039">A morphic approach to combinatorial games: the Tribonacci case</a>. RAIRO - Theoretical Informatics and Applications, 42, 2008, pp 375-393. See Table 2. [Also available from <a href="http://archive.numdam.org/item/ITA_2008__42_2_375_0">Numdam</a>]
%H A278038 V. E. Hoggatt, Jr. and Marjorie Bicknell-Johnson, <a href="https://web.archive.org/web/2024*/https://www.fq.math.ca/Scanned/20-3/hoggatt.pdf">Lexicographic Ordering and Fibonacci Representations</a>, The Fibonacci Quarterly, Vol. 20, No. 3 (1982), pp. 193-218.
%H A278038 Wolfdieter Lang, <a href="https://arxiv.org/abs/1810.09787">The Tribonacci and ABC Representations of Numbers are Equivalent</a>, arXiv preprint arXiv:1810.09787 [math.NT], 2018.
%e A278038 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.
%p A278038 # maximum index in A73 such that A73 <= n.
%p A278038 A73floorIdx := proc(n)
%p A278038 local k ;
%p A278038 for k from 3 do
%p A278038 if A000073(k) = n then
%p A278038 return k ;
%p A278038 elif A000073(k) > n then
%p A278038 return k -1 ;
%p A278038 end if ;
%p A278038 end do:
%p A278038 end proc:
%p A278038 A278038 := proc(n)
%p A278038 local k,L,nres ;
%p A278038 if n = 0 then
%p A278038 0;
%p A278038 else
%p A278038 k := A73floorIdx(n) ;
%p A278038 L := [1] ;
%p A278038 nres := n-A000073(k) ;
%p A278038 while k >= 4 do
%p A278038 k := k-1 ;
%p A278038 if nres >= A000073(k) then
%p A278038 L := [1,op(L)] ;
%p A278038 nres := nres-A000073(k) ;
%p A278038 else
%p A278038 L := [0,op(L)] ;
%p A278038 end if ;
%p A278038 end do:
%p A278038 add( op(i,L)*10^(i-1),i=1..nops(L)) ;
%p A278038 end if;
%p A278038 end proc:
%p A278038 seq(A278038(n),n=0..40) ; # _R. J. Mathar_, Jun 08 2022
%t A278038 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 *)
%Y A278038 Cf. A000073, A080843 (tribonacci word, tribonacci tree).
%Y A278038 See A003726 for the decimal representations of these binary strings.
%Y A278038 Similar sequences: A014417 (Fibonacci), A130310 (Lucas).
%K A278038 nonn,base
%O A278038 0,3
%A A278038 _N. J. A. Sloane_, Nov 16 2016