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A296063 a(n) is the smallest number (in absolute value) not yet in the sequence such that the arithmetic mean of the first n terms a(1), a(2), ..., a(n) is an integer. Preference is given to positive values of a(n); a(1)=1; 0 not allowed.

%I A296063 #21 Mar 14 2020 15:25:31
%S A296063 1,-1,3,-3,5,-5,7,-7,9,-9,11,-11,13,-13,15,-15,17,-17,19,-19,21,-21,
%T A296063 23,-23,25,-25,27,-27,29,-29,31,-31,33,-33,35,-35,37,-37,39,-39,41,
%U A296063 -41,43,-43,45,-45,47,-47,49,-49,51,-51
%N A296063 a(n) is the smallest number (in absolute value) not yet in the sequence such that the arithmetic mean of the first n terms a(1), a(2), ..., a(n) is an integer. Preference is given to positive values of a(n); a(1)=1; 0 not allowed.
%H A296063 Colin Barker, <a href="/A296063/b296063.txt">Table of n, a(n) for n = 1..1000</a>
%H A296063 <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (-1,1,1).
%F A296063 a(n) = (-1)^(n+1)*A109613(n+1). - _Michel Marcus_, Dec 05 2017
%F A296063 From _Colin Barker_, Mar 14 2020: (Start)
%F A296063 G.f.: x*(1 + x^2) / ((1 - x)*(1 + x)^2).
%F A296063 a(n) = -a(n-1) + a(n-2) + a(n-3) for n>3.
%F A296063 (End)
%t A296063 Array[(2 Floor[(# + 1)/2] - 1) (2 Boole@ OddQ@ # - 1) &, 52] (* or *)
%t A296063 Nest[Append[#, Block[{k = 1, s = 1}, While[Nand[FreeQ[#, s k], IntegerQ@ Mean[Append[#, s k]]], If[s == 1, s = -1, k++; s = 1]]; s k]] &, {1}, 51] (* _Michael De Vlieger_, Dec 12 2017 *)
%o A296063 (PARI) Vec(x*(1 + x^2) / ((1 - x)*(1 + x)^2) + O(x^50)) \\ _Colin Barker_, Mar 14 2020
%Y A296063 Cf. A193356 (partial sums), A059841 (a(n)/n), A109613.
%K A296063 sign
%O A296063 1,3
%A A296063 _Enrique Navarrete_, Dec 04 2017