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A303647 a(n) = ceiling(a(n-1)/(2^(1/3)-1)+1), a(1)=1.

%I A303647 #19 Jun 24 2018 11:45:48
%S A303647 1,5,21,82,317,1221,4699,18080,69561,267625,1029641,3961362,15240637,
%T A303647 58635641,225590199,867918160,3339160721,12846826845,49425880861,
%U A303647 190157283842,731596320957,2814686695261,10829006332499,41662675404240,160289731905481,616686228261665
%N A303647 a(n) = ceiling(a(n-1)/(2^(1/3)-1)+1), a(1)=1.
%F A303647 a(n) = A301483(n+2) - A195350(n+2) + A195339(n-2) (conjectured).
%F A303647 Conjectures from _Colin Barker_, Apr 28 2018: (Start)
%F A303647 G.f.: x*(1 + x + x^2) / ((1 - x)*(1 - 3*x - 3*x^2 - x^3)).
%F A303647 a(n) = 4*a(n-1) - 2*a(n-3) - a(n-4) for n>4.
%F A303647 (End)
%p A303647 a := proc(n) option remember;
%p A303647        if n<1 then 0 else
%p A303647        if n=1 then 1 else ceil(a(n-1)/(2^(1/3)-1)+1)
%p A303647      end if end if end proc;
%p A303647 seq(a(n), n=0..10);
%t A303647 Nest[Append[#, Ceiling[#[[-1]]/(2^(1/3) - 1) + 1]] &, {1}, 25] (* or *)
%t A303647 Rest@ CoefficientList[Series[x (1 + x + x^2)/((1 - x) (1 - 3 x - 3 x^2 - x^3)), {x, 0, 25}], x] (* _Michael De Vlieger_, Apr 28 2018 *)
%o A303647 (PARI) a(n) = if (n==1, 1, ceil(a(n-1)/(2^(1/3)-1)+1)); \\ _Michel Marcus_, Apr 28 2018
%Y A303647 Cf. A195350, A301483, A195339.
%K A303647 nonn
%O A303647 1,2
%A A303647 _Gregory Gerard Wojnar_, Apr 27 2018