|
%I #28 Sep 08 2022 08:46:15
%S -1,-2,15,74,199,414,743,1210,1839,2654,3679,4938,6455,8254,10359,
%T 12794,15583,18750,22319,26314,30759,35678,41095,47034,53519,60574,
%U 68223,76490,85399,94974,105239,116218,127935,140414,153679,167754,182663,198430,215079,232634,251119
%N a(n) = 4*n^3 - 3*n^2 - 2*n - 1.
%C 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 Primes in this sequence: 199, 743, 15583, 105239, 435359, 620999, 770239, 853079, 1738423, 3511103, 7580119, 8737039, 10006063, ...
%C If a(n) is a positive prime then n is congruent to 0 or 4 (mod 6).
%H Ilya Gutkovskiy, <a href="/A268644/a268644.pdf">Examples of the ordinary generating function for the values of cubic polynomialK</a>
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/CubicPolynomial.html">Cubic Polynomial</a>
%H <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (4,-6, 4,-1).
%F G.f.: (-1 + 2*x + 17*x^2 + 6*x^3)/(1 - x)^4.
%F a(n) = A103532(n - 1) - A005408(n), n>0.
%F a(n) = 4*a(n -1) - 6*a(n - 2) + 4*a(n - 3) - a(n - 4).
%F Sum_(n>=0} 1/a(n) = -1.407823506818026589265...
%t Table[4 n^3 - 3 n^2 - 2 n - 1, {n, 0, 40}]
%t LinearRecurrence[{4, -6, 4, -1}, {-1, -2, 15, 74}, 41]
%t CoefficientList[Series[(-1 + 2 x + 17 x^2 + 6 x^3) / (1 - x)^4, {x, 0, 40}], x] (* _Vincenzo Librandi_, Feb 10 2016 *)
%o (Magma) [4*n^3-3*n^2-2*n-1: n in [0..40]]; _Vincenzo Librandi_, Feb 10 2016
%o (PARI) a(n)=4*n^3-3*n^2-2*n-1 \\ _Charles R Greathouse IV_, Jul 26 2016
%Y Cf. A005408, A056578, A103532, A131464.
%K easy,sign
%O 0,2
%A _Ilya Gutkovskiy_, Feb 09 2016
|
