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%I #53 Mar 13 2022 19:13:00
%S 1,7,19,25,55,61,73,79,163,169,181,187,217,223,235,241,487,493,505,
%T 511,541,547,559,565,649,655,667,673,703,709,721,727,1459,1465,1477,
%U 1483,1513,1519,1531,1537,1621,1627,1639,1645,1675,1681,1693,1699
%N At step number k >= 1 the 2^(k-1) open intervals that are erased from [0,1] in the Cantor middle-third set construction are I(k,n) = (a(n)/3^k, (1+a(n))/3^k), n=1..2^(k-1).
%C Related to A005836. Gives boundaries of open intervals that have to be erased in the Cantor middle-third set construction.
%C Let g(n) = Sum_{i=0..n} (i*binomial(n+i,i)^3*binomial(n,i)^2) = A112035(n). Let b = {m>0 : g(m) != 0 (mod 3)}. Then b(n) = a(n). - Mohammed Bouayoun (bouyao(AT)wanadoo.fr), Mar 08 2004
%C Conjecture: Similarly to A191107, this increasing sequence is generated by the rules: a(1) = 1, and if x is in the sequence, then 3*x-2 and 3*x+4 are also in the sequence. - _L. Edson Jeffery_, Nov 17 2015
%H Ralf Stephan, <a href="/somedcgf.html">Some divide-and-conquer sequences ...</a>
%H Ralf Stephan, <a href="/A079944/a079944.ps">Table of generating functions</a>
%H <a href="/index/Ar#3-automatic">Index entries for 3-automatic sequences</a>.
%F a(n) = 1+6*A005836(n), n >= 1.
%F a(n) = 1+3*A005823(n), n >= 1.
%F a(n+1) = A074938(n) + A074939(n); A074938: odd numbers in A005836, A074939: even numbers in A005836. - _Philippe Deléham_, Jul 10 2005
%F Conjecture: a(n) = 2*A191107(n) - 1 = 6*A003278(n) - 5 = (a((2*n-1)*2^(k-1))+2)/3^k, k>0. - _L. Edson Jeffery_, Nov 25 2015
%e k=1: (1/3, 2/3);
%e k=2: (1/9, 2/9), (7/9, 8/9);
%e k=3: (1/27, 2/27), (7/27, 8/27), (19/27, 20/27), (25/27, 26/27); ...
%t (* (Conjectured) Choose rows large enough to guarantee that all terms < max are generated. *)
%t rows = 1000; max = 10^4; a[1] = {1}; i = 1; Do[a[n_] = {}; Do[If[1 < 3*a[n - 1][[k]] - 2 < max, AppendTo[a[n], 3*a[n - 1][[k]] - 2], Break]; If[3*a[n - 1][[k]] + 4 < max, AppendTo[a[n], 3*a[n - 1][[k]] + 4], Break], {k, Length[a[n - 1]]}]; If[a[n] == {}, Break, i++], {n, 2, 1000}]; a055246 = Take[Flatten[Table[a[n], {n, i}]], 48] (* _L. Edson Jeffery_, Nov 17 2015 *)
%t Join[{1}, 1 + 6 Accumulate[Table[(3^IntegerExponent[n, 2] + 1)/2, {n, 60}]]] (* _Vincenzo Librandi_, Nov 26 2015 *)
%o (PARI) g(n)=sum(i=0,n,i*binomial(n+i,i)^3*binomial(n,i)^2);
%o for (i=1,2000,if(Mod(g(i),3)<>0,print1(i,",")))
%o (PARI) a(n) = fromdigits(binary(n-1),3)*6 + 1; \\ _Kevin Ryde_, Apr 23 2021
%Y Cf. A003278, A005836, A005823, A055247, A112035.
%Y Cf. A191107.
%K nonn,easy
%O 1,2
%A _Wolfdieter Lang_, May 23 2000
%E Edited by _N. J. A. Sloane_, Nov 20 2015: used first comment to give more precise definition, and edited a comment at the suggestion of _L. Edson Jeffery_.
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