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Sums-complement of the Beatty sequence for 1 + phi.
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%I #27 Oct 21 2021 06:03:12

%S 1,4,9,12,17,22,25,30,33,38,43,46,51,56,59,64,67,72,77,80,85,88,93,98,

%T 101,106,111,114,119,122,127,132,135,140,145,148,153,156,161,166,169,

%U 174,177,182,187,190,195,200,203,208,211,216,221,224,229,232,237

%N Sums-complement of the Beatty sequence for 1 + phi.

%C See A276871 for a definition of sums-complement and guide to related sequences.

%C This appears to be 1 followed by A089910. - _R. J. Mathar_, Oct 05 2016

%C Mathar's conjecture is proved in the paper 'The Frobenius problem for homomorphic embeddings of languages into the integers'. See Example 1 in that paper. - _Michel Dekking_, Dec 21 2017

%D J. Shallit, "Synchronized Sequences" in Lecture Notes of Computer science 12847 pp 1-19 2021, see page 16.

%H Michel Dekking, <a href="https://arxiv.org/abs/1712.03345">The Frobenius problem for homomorphic embeddings of languages into the integers</a>, arXiv:1712.03345 [math.CO], 2017.

%H <a href="/index/Be#Beatty">Index entries for sequences related to Beatty sequences</a>

%F a(n) = 2[(n-1)phi] + n, where phi = (1+sqrt(5))/2 (see Example 1 in the paper 'The Frobenius problem for homomorphic embeddings of languages into the integers'). - _Michel Dekking_, Dec 21 2017

%F a(n) = A035336(n-1)+2 for n>1. - _Michel Dekking_, Dec 21 2017

%e The Beatty sequence for 1 + phi is A001950 = (2,5,7,10,13,15,18,20,23,26,...), with difference sequence s = A005614 + 2 = (3,2,3,3,2,3,2,3,3,...). The sums s(j)+s(j+1)+...+s(k) include (2,3,5,6,7,8,10,11,13,14,15,16,18,...), with complement (1,4,9,12,17,22,...).

%t z = 500; r = 1 + GoldenRatio; b = Table[Floor[k*r], {k, 0, z}]; (* A001950 *)

%t t = Differences[b]; (* 2 + A003849 *)

%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]; (* A276885 *)

%Y Cf. A001950, A003849, A276871.

%K nonn,easy

%O 1,2

%A _Clark Kimberling_, Oct 01 2016

%E Name edited and example corrected by _Michel Dekking_, Oct 30 2016