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A276871 Sums-complement of the Beatty sequence for sqrt(5).

%I A276871 #18 Feb 14 2024 17:27:35
%S A276871 1,10,19,28,37,48,57,66,75,86,95,104,113,124,133,142,151,162,171,180,
%T A276871 189,198,209,218,227,236,247,256,265,274,285,294,303,312,323,332,341,
%U A276871 350,359,370,379,388,397,408,417,426,435,446,455,464,473,484,493,502
%N A276871 Sums-complement of the Beatty sequence for sqrt(5).
%C A276871 The sums-complement of a sequence s(1), s(2), ... of positive integers is introduced here as the set of numbers c(1), c(2), ... such that no c(n) is a sum s(j)+s(j+1)+...+s(k) for any j and k satisfying 1 <= j <= k.  If this set is not empty, the term "sums-complement" also applies to the (possibly finite) sequence of numbers c(n) arranged in increasing order.  In particular, the difference sequence D(r) of a Beatty sequence B(r) of an irrational number r > 2 has an infinite sums-complement, abbreviated as SC(r) in the following table:
%C A276871   r                  B(r)        D(r)       SC(r)
%C A276871   ----------------------------------------------------
%C A276871   sqrt(5)            A022839     A081427    A276871
%C A276871   sqrt(6)            A022840     A276856    A276872
%C A276871   sqrt(7)            A022841     A276857    A276873
%C A276871   sqrt(8)            A022842     A276858    A276874
%C A276871   e                  A022843     A276859    A276875
%C A276871   2*e                A276853     A276860    A276876
%C A276871   Pi                 A022844     A063438    A276877
%C A276871   2*Pi               A028130     A276861    A276878
%C A276871   1+sqrt(2)          A003151     A276862    A276879
%C A276871   1+sqrt(3)          A054088     A007538    A276880
%C A276871   1+sqrt(5)          A276854     A276863    A276881
%C A276871   2+sqrt(2)          A001952     A276864    A276882
%C A276871   2+sqrt(3)          A003512     A276865    A276883
%C A276871   2+sqrt(5)          A004976     A276866    A276884
%C A276871   1+tau              A001950   2 + A003849  A276885
%C A276871   2+tau              A003231     A276867    A276886
%C A276871   3+tau              A276855     A276868    A276887
%C A276871   2+sqrt(1/2)        A182769     A276869    A276888
%C A276871   sqrt(2)+sqrt(3)    A110117     A276870    A276889
%C A276871 From _Jeffrey Shallit_, Aug 15 2023: (Start)
%C A276871 Simpler description:  this sequence represents those positive integers that CANNOT be expressed as a difference of two elements of A022839.
%C A276871 There is a 20-state Fibonacci automaton for the terms of this sequence (see a276871.pdf).  It takes as input the Zeckendorf representation of n and accepts iff n is a member of A276871. (End)
%H A276871 Luke Schaeffer, Jeffrey Shallit, and Stefan Zorcic, <a href="https://arxiv.org/abs/2402.08331">Beatty Sequences for a Quadratic Irrational: Decidability and Applications</a>, arXiv:2402.08331 [math.NT], 2024. See p. 16.
%H A276871 Jeffrey Shallit, <a href="/A276871/a276871.pdf">Fibonacci automaton for A276871</a>
%H A276871 <a href="/index/Be#Beatty">Index entries for sequences related to Beatty sequences</a>
%e A276871 The Beatty sequence for sqrt(5) is A022839 = (0,2,4,6,8,11,13,15,...), with difference sequence s = A081427 = (2,2,2,2,3,2,2,2,3,2,...).  The sums s(j)+s(j+1)+...+s(k) include (2,3,4,5,6,7,8,9,11,12,...), with complement (1,10,19,28,37,...).
%t A276871 z = 500; r = Sqrt[5]; b = Table[Floor[k*r], {k, 0, z}]; (* A022839 *)
%t A276871 t = Differences[b]; (* A081427 *)
%t A276871 c[k_, n_] := Sum[t[[i]], {i, n, n + k - 1}];
%t A276871 u[k_] := Union[Table[c[k, n], {n, 1, z - k + 1}]];
%t A276871 w = Flatten[Table[u[k], {k, 1, z}]]; Complement[Range[Max[w]], w];  (* A276871 *)
%Y A276871 Cf. A022839, A081427.
%K A276871 nonn,easy
%O A276871 1,2
%A A276871 _Clark Kimberling_, Sep 24 2016