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A276797 Partial sums of A276794.

%I A276797 #33 Oct 11 2022 14:18:17
%S A276797 0,0,1,1,1,1,2,2,2,3,3,3,3,4,4,5,5,5,5,6,6,6,7,7,7,7,8,8,8,8,9,9,9,10,
%T A276797 10,10,10,11,11,12,12,12,12,13,13,13,14,14,14,14,15,15,15,16,16,16,16,
%U A276797 17,17,18,18,18,18,19,19,19,20,20,20,20,21,21,21,21,22,22,22,23,23,23,23,24,24,25
%N A276797 Partial sums of A276794.
%C A276797 a(n+1) = z_A(n), the number of entries of A278040 (called A number in the W. Lang given there) not exceeding n, for n >= 1 and z_A(-1) := 0. - _Wolfdieter Lang_, Dec 06 2018
%C A276797 Conjecture: A140102(n) - n = a(n-1). - _N. J. A. Sloane_, Oct 26 2016 (added Mar 21 2019). This is true - see the Dekking et al. paper. - _N. J. A. Sloane_, Jul 22 2019
%H A276797 N. J. A. Sloane, <a href="/A276797/b276797.txt">Table of n, a(n) for n = 0..10000</a>
%H A276797 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 A276797 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.
%H A276797 Jeffrey Shallit, <a href="https://arxiv.org/abs/2210.03996">Some Tribonacci conjectures</a>, arXiv:2210.03996 [math.CO], 2022.
%F A276797 a(n) = Sum_{k=0..n} A276794(k).
%F A276797 a(n) = n + 1 - (A276796(n) +  A276798(n)).
%F A276797 a(n) = 2*B(n) - A(n) + 1, for n >= 0, where A(n) = A278040(n) and B(n) = A278039(n). For a proof see the W. Lang link in A278040, Proposition 7, eq. (41). - _Wolfdieter Lang_, Dec 06 2018
%p A276797 M:=12;
%p A276797 S[1]:=`0`; S[2]:=`01`; S[3]:=`0102`;
%p A276797 for n from 4 to M do S[n]:=cat(S[n-1], S[n-2], S[n-3]); od:
%p A276797 t0:=S[M]: # has 927 terms of tribonacci ternary word A080843
%p A276797 # get numbers of 0's, 1's, 2's
%p A276797 N0:=[]: N1:=[]: N2:=[]: c0:=0: c1:=0: c2:=0:
%p A276797 L:=length(t0);
%p A276797 for i from 1 to L do
%p A276797 js := substring(t0, i..i);
%p A276797 j:=convert(js,decimal,10);
%p A276797 if j=0 then c0:=c0+1; elif j=1 then c1:=c1+1; else c2:=c2+1; fi;
%p A276797 N0:=[op(N0),c0]; N1:=[op(N1),c1]; N2:=[op(N2),c2];
%p A276797 od:
%p A276797 N0; N1; N2; # prints A276796, A276797, A276798 (except A276798 is off by 1 because it does not count the initial 0 in A003146). # _N. J. A. Sloane_, Jun 08 2018
%Y A276797 Cf. A003145, A140102, A276794, A278039, A278040.
%Y A276797 A276793(n) + A276794(n) + A276791(n) = 1;
%Y A276797 A276796(n) + A276797(n) + A276798(n) = n + 1.
%K A276797 nonn,easy
%O A276797 0,7
%A A276797 _N. J. A. Sloane_, Oct 28 2016