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A288994 a(n) = n*(n+3) when n is congruent to 0 or 3 (mod 4), and n*(n+3)/2 otherwise.

%I A288994 #24 Aug 12 2022 09:22:52
%S A288994 0,2,5,18,28,20,27,70,88,54,65,154,180,104,119,270,304,170,189,418,
%T A288994 460,252,275,598,648,350,377,810,868,464,495,1054,1120,594,629,1330,
%U A288994 1404,740,779,1638,1720,902,945,1978,2068,1080,1127,2350,2448,1274,1325
%N A288994 a(n) = n*(n+3) when n is congruent to 0 or 3 (mod 4), and n*(n+3)/2 otherwise.
%H A288994 Colin Barker, <a href="/A288994/b288994.txt">Table of n, a(n) for n = 0..1000</a>
%H A288994 <a href="/index/Rec#order_09">Index entries for linear recurrences with constant coefficients</a>, signature (3,-6,10,-12,12,-10,6,-3,1).
%F A288994 a(n) = n*(n+3)/2 * (2 - floor((n+1)/2) mod 2), where n*(n+3)/2 is A000096(n).
%F A288994 a(n) = A060819(n+3)*A145979(n-2).
%F A288994 a(n) = (2*n*(n+3))/(GCD(4, n+2)*GCD(4, n+3)).
%F A288994 a(n) = A227316(n+1) - (period 4 repeat 2,1,1,2).
%F A288994 From _Colin Barker_, Jun 21 2017: (Start)
%F A288994 G.f.: x*(2 - x + 15*x^2 - 16*x^3 + 18*x^4 - 9*x^5 + 5*x^6 - 2*x^7) / ((1 - x)^3*(1 + x^2)^3).
%F A288994 a(n) = (1/8 + i/8)*(((3 - 3*i) - i*(-i)^n + i^n)*n*(3 + n)), where i=sqrt(-1). (End)
%F A288994 Sum_{n>=1} 1/a(n) = 17/18 + log(2)/6. - _Amiram Eldar_, Aug 12 2022
%t A288994 a[n_] := n (n+3) Switch[Mod[n, 4], 0|3, 1, _, 1/2]; Table[a[n], {n, 0, 50}]
%t A288994 Table[If[MemberQ[{0,3},Mod[n,4]],n(n+3),(n(n+3))/2],{n,0,50}] (* or *) LinearRecurrence[{3,-6,10,-12,12,-10,6,-3,1},{0,2,5,18,28,20,27,70,88},60] (* _Harvey P. Dale_, Jun 05 2021 *)
%o A288994 (PARI) concat(0, Vec(x*(2 - x + 15*x^2 - 16*x^3 + 18*x^4 - 9*x^5 + 5*x^6 - 2*x^7) / ((1 - x)^3*(1 + x^2)^3) + O(x^60))) \\ _Colin Barker_, Jun 21 2017
%o A288994 (PARI) i=I; a(n) = (1/8 + i/8)*(((3 - 3*i) - i*(-i)^n + i^n)*n*(3 + n)) \\ _Colin Barker_, Jun 21 2017
%Y A288994 Cf. A000096, A014695, A028552, A060819, A130658, A145979, A227316.
%K A288994 nonn,easy
%O A288994 0,2
%A A288994 _Jean-François Alcover_ and _Paul Curtz_, Jun 21 2017