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%I #18 Feb 14 2024 17:27:35
%S 1,10,19,28,37,48,57,66,75,86,95,104,113,124,133,142,151,162,171,180,
%T 189,198,209,218,227,236,247,256,265,274,285,294,303,312,323,332,341,
%U 350,359,370,379,388,397,408,417,426,435,446,455,464,473,484,493,502
%N Sums-complement of the Beatty sequence for sqrt(5).
%C 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 r B(r) D(r) SC(r)
%C ----------------------------------------------------
%C sqrt(5) A022839 A081427 A276871
%C sqrt(6) A022840 A276856 A276872
%C sqrt(7) A022841 A276857 A276873
%C sqrt(8) A022842 A276858 A276874
%C e A022843 A276859 A276875
%C 2*e A276853 A276860 A276876
%C Pi A022844 A063438 A276877
%C 2*Pi A028130 A276861 A276878
%C 1+sqrt(2) A003151 A276862 A276879
%C 1+sqrt(3) A054088 A007538 A276880
%C 1+sqrt(5) A276854 A276863 A276881
%C 2+sqrt(2) A001952 A276864 A276882
%C 2+sqrt(3) A003512 A276865 A276883
%C 2+sqrt(5) A004976 A276866 A276884
%C 1+tau A001950 2 + A003849 A276885
%C 2+tau A003231 A276867 A276886
%C 3+tau A276855 A276868 A276887
%C 2+sqrt(1/2) A182769 A276869 A276888
%C sqrt(2)+sqrt(3) A110117 A276870 A276889
%C From _Jeffrey Shallit_, Aug 15 2023: (Start)
%C Simpler description: this sequence represents those positive integers that CANNOT be expressed as a difference of two elements of A022839.
%C 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 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 Jeffrey Shallit, <a href="/A276871/a276871.pdf">Fibonacci automaton for A276871</a>
%H <a href="/index/Be#Beatty">Index entries for sequences related to Beatty sequences</a>
%e 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 z = 500; r = Sqrt[5]; b = Table[Floor[k*r], {k, 0, z}]; (* A022839 *)
%t t = Differences[b]; (* A081427 *)
%t c[k_, n_] := Sum[t[[i]], {i, n, n + k - 1}];
%t u[k_] := Union[Table[c[k, n], {n, 1, z - k + 1}]];
%t w = Flatten[Table[u[k], {k, 1, z}]]; Complement[Range[Max[w]], w]; (* A276871 *)
%Y Cf. A022839, A081427.
%K nonn,easy
%O 1,2
%A _Clark Kimberling_, Sep 24 2016
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