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A131737 Essentially even numbers followed by duplicated odd numbers.

%I A131737 #32 Jan 01 2024 23:49:52
%S A131737 0,1,1,1,2,3,3,4,5,5,6,7,7,8,9,9,10,11,11,12,13,13,14,15,15,16,17,17,
%T A131737 18,19,19,20,21,21,22,23,23,24,25,25,26,27,27,28,29,29,30,31,31,32,33,
%U A131737 33,34,35,35,36,37,37,38,39,39,40,41,41,42,43,43,44,45
%N A131737 Essentially even numbers followed by duplicated odd numbers.
%H A131737 <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (1, 0, 1, -1).
%F A131737 a(0)=0. a(1)=a(2)=1. a(3*n)=A005408(n-1). a(3*n+1)=a(3*n)+1. a(3*n+2)=a(3*n)+2, n>0.
%F A131737 O.g.f.: x*(1+x^4)/((1-x)^2*(x^2+x+1)). a(n)=(2*n-2-A057078(n))/3, n>1. - _R. J. Mathar_, Jul 16 2008
%F A131737 Euler transform of length 8 sequence [ 1, 0, 1, 1, 0, 0, 0, -1]. - _Michael Somos_, Jan 11 2011
%F A131737 0 = a(n) - a(n+1) - a(n+3) + a(n+4) if n>1. - _Michael Somos_, Nov 11 2015
%F A131737 a(n) = floor((2*n-1)/3) for n > 1. - _Werner Schulte_, Feb 27 2019
%e A131737 G.f. = x + x^2 + x^3 + 2*x^4 + 3*x^5 + 3*x^6 + 4*x^7 + 5*x^8 + 5*x^9 + 6*x^10 + ...
%p A131737 A131737 := proc(n): (1/9)*add(5*(k mod 3)+2*((k+1) mod 3)-((k+2) mod 3),k=0..n)-1+(binomial(2*n,n) mod 2)+(binomial((n+1)^2,n+3) mod 2) end: seq( A131737(n),n=0..74); # _Johannes W. Meijer_, Jun 27 2011
%t A131737 Join[{0, 1}, LinearRecurrence[{1, 0, 1, -1}, {1, 1, 2, 3}, 68]] (* _Georg Fischer_, Feb 27 2019 *)
%t A131737 Insert[Flatten[Table[If[EvenQ[n],n,{n,n}],{n,0,70}]],1,2] (* _Harvey P. Dale_, Sep 04 2020 *)
%o A131737 (PARI) {a(n) = (n==0) + (n==1) + (n\3)*2 + (n%3) - 1}; /* _Michael Somos_, Jan 11 2011 */
%Y A131737 Cf. A004396.
%K A131737 nonn,easy,less
%O A131737 0,5
%A A131737 _Paul Curtz_, Sep 19 2007
%E A131737 Edited by _R. J. Mathar_, Jul 16 2008