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A276796 Partial sums of A276793.

%I A276796 #40 Oct 11 2022 11:11:20
%S A276796 0,1,1,2,2,3,3,4,5,5,6,6,7,7,8,8,9,9,10,10,11,12,12,13,13,14,14,15,15,
%T A276796 16,16,17,18,18,19,19,20,20,21,21,22,22,23,23,24,25,25,26,26,27,27,28,
%U A276796 29,29,30,30,31,31,32,32,33,33,34,34,35,36,36,37,37,38,38,39,39,40,40,41,42,42,43,43,44
%N A276796 Partial sums of A276793.
%C A276796 a(n+1) = z_B(n), the number of B numbers A278039 not exceeding n. - _Wolfdieter Lang_, Dec 05 2018
%C A276796 Conjecture: A140100(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 A276796 N. J. A. Sloane, <a href="/A276796/b276796.txt">Table of n, a(n) for n = 0..10000</a>
%H A276796 F. Michel Dekking, Jeffrey Shallit, and N. J. A. Sloane, <a href="https://doi.org/10.37236/8905">Queens in exile: non-attacking queens on infinite chess boards</a>, Electronic J. Combin., 27:1 (2020), #P1.52.
%H A276796 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 A276796 Jeffrey Shallit, <a href="https://arxiv.org/abs/2210.03996">Some Tribonacci conjectures</a>, arXiv:2210.03996 [math.CO], 2022.
%F A276796 a(n) = Sum_{k=0..n} A276793(k), n >= 0.
%F A276796 a(n) = n + 1 - (A276797(n) + A276798(n)).
%F A276796 a(n) = A(n) - B(n) - (n + 1), where A(n) = A278040(n) and B(n) = A278039(n), n >= 0. For a proof see the W. Lang link in A278040, Proposition 7, eq. (42). - _Wolfdieter Lang_, Dec 05 2018
%p A276796 M:=12;
%p A276796 S[1]:=`0`; S[2]:=`01`; S[3]:=`0102`;
%p A276796 for n from 4 to M do S[n]:=cat(S[n-1], S[n-2], S[n-3]); od:
%p A276796 t0:=S[M]: # has 927 terms of tribonacci ternary word A080843
%p A276796 # get numbers of 0's, 1's, 2's
%p A276796 N0:=[]: N1:=[]: N2:=[]: c0:=0: c1:=0: c2:=0:
%p A276796 L:=length(t0);
%p A276796 for i from 1 to L do
%p A276796 js := substring(t0, i..i);
%p A276796 j:=convert(js,decimal,10);
%p A276796 if j=0 then c0:=c0+1; elif j=1 then c1:=c1+1; else c2:=c2+1; fi;
%p A276796 N0:=[op(N0),c0]; N1:=[op(N1),c1]; N2:=[op(N2),c2];
%p A276796 od:
%p A276796 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 A276796 Cf. A003144, A140100, A276793 (first differences), A278039, A278040.
%Y A276796 A276793(n) + A276794(n) + A276791(n) = 1;
%Y A276796 A276796(n) + A276797(n) + A276798(n) = n + 1.
%K A276796 nonn,easy
%O A276796 0,4
%A A276796 _N. J. A. Sloane_, Oct 28 2016