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A268644 a(n) = 4*n^3 - 3*n^2 - 2*n - 1.

%I A268644 #40 Feb 16 2025 08:33:30
%S A268644 -1,-2,15,74,199,414,743,1210,1839,2654,3679,4938,6455,8254,10359,
%T A268644 12794,15583,18750,22319,26314,30759,35678,41095,47034,53519,60574,
%U A268644 68223,76490,85399,94974,105239,116218,127935,140414,153679,167754,182663,198430,215079,232634,251119
%N A268644 a(n) = 4*n^3 - 3*n^2 - 2*n - 1.
%C A268644 In general, the ordinary generating function for the values of cubic polynomial p*n^3 + q*n^2 + k*n + m is (m + (p + q + k - 3*m)*x + (4*p - 2*k + 3*m)*x^2 + (p - q + k - m)*x^3)/(1 - x)^4.
%C A268644 Primes in this sequence: 199, 743, 15583, 105239, 435359, 620999, 770239, 853079, 1738423, 3511103, 7580119, 8737039, 10006063, ...
%C A268644 If a(n) is a positive prime then n is congruent to 0 or 4 (mod 6).
%H A268644 Ilya Gutkovskiy, <a href="/A268644/a268644.pdf">Examples of the ordinary generating function for the values of cubic polynomialK</a>
%H A268644 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/CubicPolynomial.html">Cubic Polynomial</a>
%H A268644 <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (4,-6,4,-1).
%F A268644 G.f.: (-1 + 2*x + 17*x^2 + 6*x^3)/(1 - x)^4.
%F A268644 a(n) = A103532(n - 1) - A005408(n), n>0.
%F A268644 a(n) = 4*a(n - 1) - 6*a(n - 2) + 4*a(n - 3) - a(n - 4).
%F A268644 Sum_{n>=0} 1/a(n) = -1.407823506818026589265...
%F A268644 E.g.f.: exp(x)*(-1 - x + 9*x^2 + 4*x^3). - _Stefano Spezia_, Nov 17 2024
%t A268644 Table[4 n^3 - 3 n^2 - 2 n - 1, {n, 0, 40}]
%t A268644 LinearRecurrence[{4, -6, 4, -1}, {-1, -2, 15, 74}, 41]
%t A268644 CoefficientList[Series[(-1 + 2 x + 17 x^2 + 6 x^3) / (1 - x)^4, {x, 0, 40}], x] (* _Vincenzo Librandi_, Feb 10 2016 *)
%o A268644 (Magma) [4*n^3-3*n^2-2*n-1: n in [0..40]]; // _Vincenzo Librandi_, Feb 10 2016
%o A268644 (PARI) a(n)=4*n^3-3*n^2-2*n-1 \\ _Charles R Greathouse IV_, Jul 26 2016
%Y A268644 Cf. A005408, A056578, A103532, A131464.
%K A268644 easy,sign
%O A268644 0,2
%A A268644 _Ilya Gutkovskiy_, Feb 09 2016