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A024219 a(n) = floor( (2nd elementary symmetric function of S(n))/(first elementary symmetric function of S(n)) ), where S(n) = {first n+1 positive integers congruent to 1 mod 3}.

%I A024219 #29 May 20 2019 10:26:13
%S A024219 0,3,7,12,19,28,38,49,62,77,93,110,129,150,172,195,220,247,275,304,
%T A024219 335,368,402,437,474,513,553,594,637,682,728,775,824,875,927,980,1035,
%U A024219 1092,1150,1209,1270,1333,1397,1462,1529,1598,1668,1739,1812,1887,1963,2040
%N A024219 a(n) = floor( (2nd elementary symmetric function of S(n))/(first elementary symmetric function of S(n)) ), where S(n) = {first n+1 positive integers congruent to 1 mod 3}.
%H A024219 Andrew Howroyd, <a href="/A024219/b024219.txt">Table of n, a(n) for n = 1..1000</a>
%H A024219 Wikipedia, <a href="https://en.wikipedia.org/wiki/Elementary_symmetric_polynomial">Elementary symmetric polynomial</a>
%H A024219 <a href="/index/Rec#order_05">Index entries for linear recurrences with constant coefficients</a>, signature (3,-4,4,-3,1).
%F A024219 From _R. J. Mathar_, Oct 08 2011: (Start)
%F A024219 Conjecture: a(n) = 3*a(n-1) - 4*a(n-2) + 4*a(n-3) - 3*a(n-4) + a(n-5);
%F A024219 g.f.: x^2*(-3+2*x-3*x^2+x^3) / ( (x^2+1)*(x-1)^3 ). (End)
%F A024219 From _Andrew Howroyd_, Aug 12 2018: (Start)
%F A024219 The above conjectures are true.
%F A024219 a(n) = floor(A024212(n) / A000326(n+1)).
%F A024219 a(n) = floor(n*(9*n^2 + 9*n - 2)/(4*(3*n + 2))).
%F A024219 (End)
%t A024219 LinearRecurrence[{3,-4,4,-3,1},{0,3,7,12,19},60] (* _Harvey P. Dale_, May 20 2019 *)
%o A024219 (PARI) a(n)=floor(sum(j=0, n, sum(k=j+1, n, (3*j+1)*(3*k+1)))/sum(i=0, n, (3*i+1))) \\ _Andrew Howroyd_, Aug 12 2018
%o A024219 (PARI) a(n) = floor(n*(9*n^2+9*n-2)/(4*(3*n+2))); \\ _Andrew Howroyd_, Aug 12 2018
%Y A024219 Cf. A000326, A024212,
%K A024219 nonn,easy
%O A024219 1,2
%A A024219 _Clark Kimberling_
%E A024219 More terms from _Joshua Zucker_, May 20 2006