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A127218 Half-indexed Lucas numbers second version L(n)=A000032=Lucas numbers a(0)=2, a(1)=2, a(2)=1, a(3)=2, a(4)=3, a(5)=3, a(2n)=L(n), for n>2: a(2n+1)=L(n)+L(n-3)=2*L(n-1) for n>5: a(n)+a(n+2)=a(n+4) a(2n)=L(n), so a(n)=L(n/2).

%I A127218 #13 Jun 22 2022 14:34:26
%S A127218 2,2,1,2,3,3,4,6,7,8,11,14,18,22,29,36,47,58,76,94,123,152,199,246,
%T A127218 322,398,521,644,843,1042,1364,1686,2207,2728,3571,4414,5778,7142,
%U A127218 9349,11556,15127,18698,24476,30254,39603,48952,64079
%N A127218 Half-indexed Lucas numbers second version L(n)=A000032=Lucas numbers a(0)=2, a(1)=2, a(2)=1, a(3)=2, a(4)=3, a(5)=3, a(2n)=L(n), for n>2: a(2n+1)=L(n)+L(n-3)=2*L(n-1) for n>5: a(n)+a(n+2)=a(n+4) a(2n)=L(n), so a(n)=L(n/2).
%C A127218 b(n)=A096748(n-1): for n>5: b(n)+b(n+4)=a(n+2) for n>5: a(n)+a(n+4)=5*b(n+2).
%H A127218 Colin Barker, <a href="/A127218/b127218.txt">Table of n, a(n) for n = 0..1000</a>
%H A127218 <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (0,1,0,1).
%F A127218 From _Colin Barker_, Aug 03 2020: (Start)
%F A127218 G.f.: (1 + x)*(2 - x^2 + x^3 - x^4 + x^7 - x^8) / (1 - x^2 - x^4).
%F A127218 a(n) = a(n-2) + a(n-4) for n>10.
%F A127218 (End)
%p A127218 b[0]:=2:b[1]:=1:for n from 2 to 80 do b[n]:=b[n-1]+b[n-2] od: a[0]:=2:a[1]:=2:a[2]:=1:a[3]:=2:a[4]:=3:a[5]:=3: for n from 3 to 39 do a[2*n]:=b[n]:a[2*n+1]:=b[n]+b[n-3] od: seq(a[n],n=0..79);
%t A127218 LinearRecurrence[{0,1,0,1},{2,2,1,2,3,3,4,6,7,8},60] (* _Harvey P. Dale_, Jun 22 2022 *)
%o A127218 (PARI) Vec((1 + x)*(2 - x^2 + x^3 - x^4 + x^7 - x^8) / (1 - x^2 - x^4) + O(x^45)) \\ _Colin Barker_, Aug 03 2020
%Y A127218 Cf. A000032, A096748.
%K A127218 easy,nonn
%O A127218 0,1
%A A127218 _Miklos Kristof_, Mar 28 2007