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A140102 Term-by-term differences of A140101 and A140100; also, equals the complement of A140103, which is the term-by-term sums of A140101 and A140100, where A140101 is the complement of A140100.

%I A140102 #39 Mar 07 2020 13:50:20
%S A140102 0,1,2,4,5,6,7,9,10,11,13,14,15,16,18,19,21,22,23,24,26,27,28,30,31,
%T A140102 32,33,35,36,37,38,40,41,42,44,45,46,47,49,50,52,53,54,55,57,58,59,61,
%U A140102 62,63,64,66,67,68,70,71,72,73,75,76,78,79,80,81,83,84,85,87,88,89,90,92,93
%N A140102 Term-by-term differences of A140101 and A140100; also, equals the complement of A140103, which is the term-by-term sums of A140101 and A140100, where A140101 is the complement of A140100.
%H A140102 N. J. A. Sloane, <a href="/A140102/b140102.txt">Table of n, a(n) for n = 0..50000</a>, Sep 13 2016 (First 1001 terms from Reinhard Zumkeller)
%H A140102 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 A140102 N. J. A. Sloane, <a href="/A140100/a140100.txt">Maple program for A140100, A140101, A140102, A140103</a>
%F A140102 a(n) = A140101(n) - A140100(n).
%F A140102 Theorem: the limit of A140103(n)/A140102(n) = t^2 = 3.38297576...
%F A140102 where the limit of A140101(n)/A140100(n) = t = 1.839286755...
%F A140102 and t = tribonacci constant satisfies: t^3 = 1 + t + t^2.
%p A140102 See link.
%t A140102 nmax = 100; y[0] = 0; x[1] = 1; y[1] = 2; x[n_] := x[n] = For[yn = y[n-1] + 1, True, yn++, For[xn = x[n-1] + 1, xn < yn, xn++, xx = Array[x, n-1]; yy = Array[y, n-1]; If[FreeQ[xx, xn | yn] && FreeQ[yy, xn | yn] && FreeQ[yy - xx, yn - xn] && FreeQ[yy + xx, yn - xn], y[n] = yn; Return[xn]]]];
%t A140102 Do[x[n], {n, 1, nmax}];
%t A140102 Join[{0}, yy - xx] (* _Jean-François Alcover_, Aug 01 2018 *)
%o A140102 (PARI) {X=[1];Y=[2];D=[1];S=[3];print1(Y[1]-X[1]","); for(n=1,100,for(j=2,2*n,if(setsearch(Set(concat(X,Y)),j)==0,Xt=concat(X,j); for(k=j+1,3*n,if(setsearch(Set(concat(Xt,Y)),k)==0, if(setsearch(Set(concat(D,S)),k-j)==0,if(setsearch(Set(concat(D,S)),k+j)==0, X=Xt;Y=concat(Y,k);D=concat(D,k-j);S=concat(S,k+j); print1(Y[ #X]-X[ #Y]",");break);break))))))}
%Y A140102 Cf. A140103 (complement); A140100, A140101; A058265.
%Y A140102 For first differences of A140100, A140101, A140102, A140103 see A305392, A305374, A305393, A305394.
%K A140102 nonn
%O A140102 0,3
%A A140102 _Paul D. Hanna_, Jun 04 2008
%E A140102 Terms computed by _Reinhard Zumkeller_.
%E A140102 Offset and initial term changed by _N. J. A. Sloane_, Oct 10 2016