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%I #31 Sep 08 2022 08:46:03
%S 1,1,1,3,3,5,3,7,5,9,5,11,7,13,7,15,9,17,9,19,11,21,11,23,13,25,13,27,
%T 15,29,15,31,17,33,17,35,19,37,19,39,21,41,21,43,23,45,23,47,25,49,25,
%U 51,27,53,27,55,29,57,29,59,31,61,31,63,33,65,33,67,35,69,35,71,37,73,37,75,39,77,39,79,41,81,41,83,43,85,43
%N a(4*n) = a(4*n+2) = a(2*n+1) = 2*n + 1.
%C A214282(n) and -A214283(n) are companions. Separately or together, they have many links with the Catalan's numbers A000108(n). Examples:
%C A214282(n+1) - 2*A214282(n) = -1, -1, 1, 0, -2, -5, 5, 0, -14, -42, 42, 0, -132, ....
%C 2*A214283(n) - A214283(n+1) = 1, 0, -1, -2, 2, 0, -5, -14, 14, 0, -42, -132, 132, ....
%C A214282(n) + A214283(n) = 1, 0, -1, 0, 2, 0, -5, 0, 14, 0, -42,... (A126120).
%C The companion to a(n) is b(n) = -A214283(n)/(1,1,1,1,2,2,5,5,...) = 0, 1, 2, 3, 2, 5, 4, 7, 4, 9, 6, ....
%C a(n) - b(n) = A056594(n).
%C Discovered as a(n) = A214282(n+1)/A000108([n/2]). See abs(A129996(n-2)).
%H G. C. Greubel, <a href="/A215495/b215495.txt">Table of n, a(n) for n = 0..1000</a>
%H <a href="/index/Rec#order_06">Index entries for linear recurrences with constant coefficients</a>, signature (0,1,0,1,0,-1).
%F a(n+3) = (A185048(n+3)=2,2,4,2,... ) + 1.
%F a(n+2) - a(n) = 0, 2, 2, 2. (Period 4).
%F a(n) = 2*a(n-4) - a(n-8).
%F a(2*n) = A109613(n).
%F a(n+1) - a(n) = 2* (-1)^n * A059169(n).
%F G.f. : ( 1+x+2*x^3+x^4+x^5 ) / ( (x^2+1)*(x-1)^2*(1+x)^2 ). - _Jean-François Alcover_, Aug 14 2012
%t a[n_?EvenQ] := n/2 + Boole[Mod[n, 4] == 0]; a[n_?OddQ] := n; Table[a[n], {n, 0, 86}] (* _Jean-François Alcover_, Aug 14 2012 *)
%t LinearRecurrence[{0,1,0,1,0,-1}, {1,1,1,3,3,5}, 50] (* _G. C. Greubel_, Apr 23 2018 *)
%o (PARI) x='x+O('x^30); Vec(( 1+x+2*x^3+x^4+x^5 )/( (x^2+1)*(x-1)^2*(1+x)^2 )) \\ _G. C. Greubel_, Apr 23 2018
%o (Magma) I:=[1,1,1,3,3,5]; [n le 6 select I[n] else Self(n-2) +Self(n-4) -Self(n-6): n in [1..30]]; // _G. C. Greubel_, Apr 23 2018
%Y Cf. A130823, A090192.
%K nonn,easy
%O 0,4
%A _Paul Curtz_, Aug 13 2012
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