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%I #47 Dec 31 2023 11:27:40
%S 0,2,4,6,4,6,8,10,8,10,12,14,12,14,16,18,16,18,20,22,20,22,24,26,24,
%T 26,28,30,28,30,32,34,32,34,36,38,36,38,40,42,40,42,44,46,44,46,48,50,
%U 48,50,52,54,52,54,56,58,56,58,60,62,60,62,64,66,64,66,68,70,68,70,72,74
%N a(n) = n + (n mod 4).
%H Stefano Spezia, <a href="/A083220/b083220.txt">Table of n, a(n) for n = 0..10000</a>
%H <a href="/index/Rec#order_05">Index entries for linear recurrences with constant coefficients</a>, signature (1,0,0,1,-1).
%F a(n) = 2*A083219(n).
%F a(n) = a(n-1) + 2*(n mod 2 + (n mod 4 -1)*(1- n mod 2)), a(0)=0.
%F a(n) = (3 - (-1)^n - (1+i)*(-i)^n - (1-i)*i^n + 2*n)/2 where i=sqrt(-1). - _Colin Barker_, Oct 13 2014
%F G.f.: -2*x*(x^3-x^2-x-1) / ((x-1)^2*(x+1)*(x^2+1)). - _Colin Barker_, Oct 13 2014
%F For n > 4, a(n) = a(n-4) + 4. - _Zak Seidov_, Feb 23 2017
%F G.f.: 1/(1-x)^2 + 1/(2*(1-x)) - 1/(2*(1+x)) - (1+x)/(1+x^2). - _Michael Somos_, Feb 23 2017
%F E.g.f.: (1 + x)*cosh(x) - cos(x) + (2 + x)*sinh(x) - sin(x). - _Stefano Spezia_, May 28 2021
%F Sum_{n>=1} (-1)^(n+1)/a(n) = log(2) - 1/2 (A187832). - _Amiram Eldar_, Aug 21 2023
%e G.f. = 2*x + 4*x^2 + 6*x^3 + 4*x^4 + 6*x^5 + 8*x^6 + 10*x^7 + 8*x^8 + 10*x^9 + ...
%t a[n_] := Mod[n, 4] + n; (* _Michael Somos_, Feb 23 2017 *)
%o (PARI) concat(0, Vec(-2*x*(x^3-x^2-x-1)/((x-1)^2*(x+1)*(x^2+1)) + O(x^100))) \\ _Colin Barker_, Oct 13 2014
%o (PARI) {a(n) = n%4 + n}; /* _Michael Somos_, Feb 23 2017 */
%Y Cf. A010873 (n mod 4), A083219, A187832.
%K nonn,easy
%O 0,2
%A _Reinhard Zumkeller_, Apr 22 2003
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