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%I #23 Jul 06 2020 15:13:13
%S 1,2,5,6,9,12,13,16,17,20,23,24,27,30,31,34,35,38,41,42,45,46,49,52,
%T 53,56,59,60,63,64,67,70,71,74,77,78,81,82,85,88,89,92,93,96,99,100,
%U 103,106,107,110,111,114,117,118,121,122,125,128,129,132,135,136
%N Sums-complement of the Beatty sequence for 2 + phi.
%C See A276871 for a definition of sums-complement and guide to related sequences.
%C From _Michel Dekking_, Apr 30 2019: (Start)
%C This sequence is a generalized Beatty sequence. According to Theorem 3.2 in the paper "The Frobenius problem for homomorphic embeddings of languages into the integers" this sequence (as a subset of the natural numbers) is the complement of the union of the two Beatty sequences
%C V := A003231 and W = V+1 (as subsets of the natural numbers) given by
%C V(n):= A(n)+2n = 3,7,10,14,..., W(n):=A(n)+2n+1 = 4,8,11,15,...
%C Here A = A000201, the lower Wythoff sequence.
%C Since the sequence Delta A = A014675 of first differences of A is the infinite Fibonacci word on the alphabet {2,1}, the sequence Delta V = (V(n+1)-V(n)) is the infinite Fibonacci word on the alphabet {4,3}. (Delta V equals A276867 shifted by 1.)
%C Now if for some k, Delta V(k) = 4, then a distance 3 plus a distance 1 are generated between three consecutive numbers in the complement, whereas if Delta V(k) = 3, then only a distance 3 is generated between two consecutive numbers in the complement.
%C This means that (skipping a(1)=1)
%C Delta a = (a(n+1)-a(n)) = gamma(Delta V),
%C where gamma is the morphism
%C gamma(4) = 31, gamma(3) = 3.
%C Since the Fibonacci word is a fixed point of the morphism 0->01, 1->0, this implies that Delta a, skipping a(1)=1, is the Fibonacci word on the alphabet {3,1}. It follows that
%C a(n+1) = 2*A(n) - n + 1.
%C (End)
%H J.-P. Allouche, F. M. Dekking, <a href="https://arxiv.org/abs/1809.03424">Generalized Beatty sequences and complementary triples</a>, arXiv:1809.03424 [math.NT], 2018.
%H Michel Dekking, <a href="https://www.sciencedirect.com/science/article/pii/S0304397518302573">The Frobenius problem for homomorphic embeddings of languages into the integers</a>, Theoretical Computer Science 732, 7 July 2018, 73-79.
%H <a href="/index/Be#Beatty">Index entries for sequences related to Beatty sequences</a>
%F a(n) = 2*floor((n-1)*phi) - n + 2, where phi is the golden mean.
%e The Beatty sequence for 2 + phi is 0 followed by A003231, which is (0,3,7,10,14,18,21,...), with difference sequence s = A276867 = (3,4,3,4,4,3,4,3,4,4,3,4,4,3,4,3,4,4,3,...). The sums s(j)+s(j+1)+...+s(k) include (3,4,7,8,10,12,14,15,...), with complement (1,2,5,6,9,12,13,16,...).
%t z = 500; r = 2 + GoldenRatio; b = Table[Floor[k*r], {k, 0, z}]; (* A003231 *)
%t t = Differences[b]; (* A276867 *)
%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]; (* A276886 *)
%Y Cf. A003231, A276867, A276871, A296184(2+phi).
%K nonn,easy
%O 1,2
%A _Clark Kimberling_, Oct 01 2016
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