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%I #72 Sep 08 2022 08:46:20
%S 1,3,3,5,7,9,9,11,13,15,15,17,19,21,21,23,25,27,27,29,31,33,33,35,37,
%T 39,39,41,43,45,45,47,49,51,51,53,55,57,57,59,61,63,63,65,67,69,69,71,
%U 73,75,75,77,79,81,81,83,85,87,87,89,91,93,93
%N a(n) = (3*n + ((n-2) mod 4))/2.
%C The product (2/3) * (4/3) * (6/5) * (6/7) * (8/9) * (10/9) * (12/11) * (12/13) * ... = Pi/(2*sqrt(3)). The denominators are a(n) for n >= 1 and numerators are a(n-1) + A093148(n) for n >= 1 -> [2, 4, 6, 6, 8, 10, 12, 12, ...].
%C Let r(n) = (a(n)-1)/(a(n)+1)) if a(n) mod 4 = 1, (a(n)+1)/(a(n)-1)) otherwise; then Product_{n>=1} r(n) = (2/1) * (2/1) * (2/3) * (4/3) * (4/5) * (4/5) * (6/5) * (6/7) * ... = Pi*sqrt(3)/2 = 2.72069904635132677...
%C The odd numbers of partial sums this sequence, are identified with the A003215 sequence. Also the prime numbers that appear in partial sums in this sequence, are identified with the A002407 sequence.
%H G. C. Greubel, <a href="/A293990/b293990.txt">Table of n, a(n) for n = 0..5000</a>
%H <a href="/index/Rec#order_05">Index entries for linear recurrences with constant coefficients</a>, signature (1,0,0,1,-1).
%F Sum_{n>=0} 1/a(n)^2 = 5*Pi^2/36 = 1.3707783890401886970... = 10*A086729.
%F (a(n) - n) * (-1)^(n+1) = A134967(n) for n >= 0.
%F a(n) - n = A162330(n) for n >= 0.
%F a(n) - n = A285869(n+1) for n >= 0.
%F a(n) + a(n+1) = A157932(n+2) for n >= 0.
%F a(n) + (2*n+1) = A047298(n+1) for n >= 0.
%F From _Colin Barker_, Oct 21 2017: (Start)
%F G.f.: x*(1 + 2*x + 2*x^3 + x^4) / ((1 - x)^2*(1 + x)*(1 + x^2)).
%F a(n) = a(n-1) + a(n-4) - a(n-5) for n > 5.
%F (End)
%F a(n + 8) = a(n) + 12. - _David A. Corneth_, Oct 21 2017
%F a(4*k+4) * a(4*k+3) - a(4*k+2) * a(4*k+1) = 2*A063305(k+3) for k >= 0.
%F Sum_{n>=0} 1/(a(n) + a(n+2))^2 = (4*Pi^2 - 27) / 108 = (A214549 - 1) / 4.
%p A293990:=n->(3*n+((n-2) mod 4))/2: seq(A293990(n), n=0..100); # _Wesley Ivan Hurt_, Oct 29 2017
%t Table[(3*n + Mod[(n - 2), 4])/2, {n, 0, 100}] (* _Wesley Ivan Hurt_, Oct 29 2017 *)
%t f[n_] := (3n + Mod[n - 2, 4])/2; Array[f, 65, 0] (* or *)
%t LinearRecurrence[{1, 0, 0, 1, -1}, {1, 3, 3, 5, 7}, 65] (* or *)
%t CoefficientList[ Series[(x^4 + 2x^3 + 2x + 1)/((x - 1)^2 (x^3 + x^2 + x + 1)), {x, 0, 64}], x] (* _Robert G. Wilson v_, Nov 28 2017 *)
%o (PARI) a(n) = (3*n + (n-2)%4) / 2
%o (PARI) Vec(x*(1 + 2*x + 2*x^3 + x^4) / ((1 - x)^2*(1 + x)*(1 + x^2)) + O(x^30)) \\ _Colin Barker_, Oct 21 2017
%o (PARI) first(n) = my(start=[1,3,3,5,7,9,9,11]); if(n<=8, return(start)); my(res=vector(n)); for (i=1, 8, res[i] = start[i]); for(i = 1, n-8 ,res[i+8] = res[i] + 12); res \\ _David A. Corneth_, Oct 21 2017
%o (Magma) [(3*n+((n-2) mod 4))/2 : n in [0..100]]; // _Wesley Ivan Hurt_, Oct 29 2017
%Y Cf. A007310, A047298, A063305, A093148, A134967, A157932, A162330, A168329, A214549, A285869, A003215, A002407.
%K nonn,easy
%O 0,2
%A _Dimitris Valianatos_, Oct 21 2017
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