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A276873 Sums-complement of the Beatty sequence for sqrt(7).

%I A276873 #10 Feb 14 2024 17:27:42
%S A276873 1,4,9,12,17,20,25,28,33,36,41,46,49,54,57,62,65,70,73,78,81,86,91,94,
%T A276873 99,102,107,110,115,118,123,128,131,136,139,144,147,152,155,160,163,
%U A276873 168,173,176,181,184,189,192,197,200,205,208,213,218,221,226,229
%N A276873 Sums-complement of the Beatty sequence for sqrt(7).
%C A276873 See A276871 for a definition of sums-complement and guide to related sequences.
%H A276873 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 A276873 <a href="/index/Be#Beatty">Index entries for sequences related to Beatty sequences</a>
%e A276873 The Beatty sequence for sqrt(7) is A022841 = (0,2,5,7,10,13,...), with difference sequence s = A276857 = (2,3,2,3,3,2,3,3,2,3,3,2,3,3,2,...).  The sums s(j)+s(j+1)+...+s(k) include (2,3,6,7,8,10,11,13,...), with complement (1,4,9,12,17,...).
%t A276873 z = 500; r = Sqrt[7]; b = Table[Floor[k*r], {k, 0, z}]; (* A022841 *)
%t A276873 t = Differences[b]; (* A276857 *)
%t A276873 c[k_, n_] := Sum[t[[i]], {i, n, n + k - 1}];
%t A276873 u[k_] := Union[Table[c[k, n], {n, 1, z - k + 1}]];
%t A276873 w = Flatten[Table[u[k], {k, 1, z}]]; Complement[Range[Max[w]], w]  (* A276873 *)
%Y A276873 Cf. A022841, A276857, A276871.
%K A276873 nonn,easy
%O A276873 1,2
%A A276873 _Clark Kimberling_, Sep 27 2016