This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A293990 #75 Nov 23 2024 18:16:45
%S A293990 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 A293990 39,39,41,43,45,45,47,49,51,51,53,55,57,57,59,61,63,63,65,67,69,69,71,
%U A293990 73,75,75,77,79,81,81,83,85,87,87,89,91,93,93
%N A293990 a(n) = (3*n + ((n-2) mod 4))/2.
%C A293990 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 A293990 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 A293990 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 A293990 G. C. Greubel, <a href="/A293990/b293990.txt">Table of n, a(n) for n = 0..5000</a>
%H A293990 <a href="/index/Rec#order_05">Index entries for linear recurrences with constant coefficients</a>, signature (1,0,0,1,-1).
%F A293990 Sum_{n>=0} 1/a(n)^2 = 5*Pi^2/36 = 1.3707783890401886970... = 10*A086729.
%F A293990 (a(n) - n) * (-1)^(n+1) = A134967(n) for n >= 0.
%F A293990 a(n) - n = A162330(n) for n >= 0.
%F A293990 a(n) - n = A285869(n+1) for n >= 0.
%F A293990 a(n) + a(n+1) = A157932(n+2) for n >= 0.
%F A293990 a(n) + (2*n+1) = A047298(n+1) for n >= 0.
%F A293990 From _Colin Barker_, Oct 21 2017: (Start)
%F A293990 G.f.: x*(1 + 2*x + 2*x^3 + x^4) / ((1 - x)^2*(1 + x)*(1 + x^2)).
%F A293990 a(n) = a(n-1) + a(n-4) - a(n-5) for n > 5.
%F A293990 (End)
%F A293990 a(n + 8) = a(n) + 12. - _David A. Corneth_, Oct 21 2017
%F A293990 a(4*k+4) * a(4*k+3) - a(4*k+2) * a(4*k+1) = 2*A063305(k+3) for k >= 0.
%F A293990 Sum_{n>=0} 1/(a(n) + a(n+2))^2 = (4*Pi^2 - 27) / 108 = (A214549 - 1) / 4.
%p A293990 A293990:=n->(3*n+((n-2) mod 4))/2: seq(A293990(n), n=0..100); # _Wesley Ivan Hurt_, Oct 29 2017
%t A293990 Table[(3*n + Mod[(n - 2), 4])/2, {n, 0, 100}] (* _Wesley Ivan Hurt_, Oct 29 2017 *)
%t A293990 f[n_] := (3n + Mod[n - 2, 4])/2; Array[f, 65, 0] (* or *)
%t A293990 LinearRecurrence[{1, 0, 0, 1, -1}, {1, 3, 3, 5, 7}, 65] (* or *)
%t A293990 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 A293990 (PARI) a(n) = (3*n + (n-2)%4) / 2
%o A293990 (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 A293990 (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 A293990 (Magma) [(3*n+((n-2) mod 4))/2 : n in [0..100]]; // _Wesley Ivan Hurt_, Oct 29 2017
%Y A293990 Cf. A007310, A047298, A063305, A093148, A134967, A157932, A162330, A168329, A214549, A285869, A003215, A002407.
%K A293990 nonn,easy
%O A293990 0,2
%A A293990 _Dimitris Valianatos_, Oct 21 2017