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%I #31 May 22 2021 04:38:37
%S 4,18,48,100,180,294,448,648,900,1210,1584,2028,2548,3150,3840,4624,
%T 5508,6498,7600,8820,10164,11638,13248,15000,16900,18954,21168,23548,
%U 26100,28830,31744,34848,38148,41650,45360,49284,53428,57798,62400
%N a(n) = n*(n+1)^2.
%C Former name was "Numbers k such that k*x^3 + x + 1 is not prime."
%C Theorem: y = k*x^3 + x + 1 is not prime for k = 4, 18, 48, ..., n*(n+1)^2. Proof: n*(n+1)^2*x^3 + x + 1 = ((n+1)*x + 1)*((n^2+n)*x^2 - n*x + 1). Thus (n+1)*x + 1 divides y. This could possibly be used as a pre-test for compositeness. This sequence is the same as beginning with the third term of A045991.
%H Jinyuan Wang, <a href="/A114364/b114364.txt">Table of n, a(n) for n = 1..1000</a>
%H <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (4,-6,4,-1).
%F a(n) = n*(n+1)^2.
%F G.f.: 2 * (2 + x)/(-1 + x)^4. - _Michael De Vlieger_, Feb 03 2019
%F From _Amiram Eldar_, Jan 02 2021: (Start)
%F Sum_{n>=1} 1/a(n) = 2 - Pi^2/6.
%F Sum_{n>=1} (-1)^(n+1)/a(n) = Pi^2/12 +2*log(2) - 2. (End)
%F E.g.f.: exp(x)*x*(4 + 5*x + x^2). - _Stefano Spezia_, May 20 2021
%p seq(2*binomial(n,2)*n, n=2..40); # _Zerinvary Lajos_, Apr 25 2007
%t CoefficientList[Series[(2 (2 + x))/(-1 + x)^4, {x, 0, 38}], x] (* or *)
%t Array[# (# + 1)^2 &, 39] (* _Michael De Vlieger_, Feb 03 2019 *)
%o (PARI) g2(n) = for(x=1,n,y=x*(x+1)^2;print1(y","))
%Y Cf. A045991.
%Y Equals twice A006002.
%K easy,nonn
%O 1,1
%A _Cino Hilliard_, Feb 09 2006
%E Name changed by _Jon E. Schoenfield_, Feb 03 2019
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