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

-1, -2, 15, 74, 199, 414, 743, 1210, 1839, 2654, 3679, 4938, 6455, 8254, 10359, 12794, 15583, 18750, 22319, 26314, 30759, 35678, 41095, 47034, 53519, 60574, 68223, 76490, 85399, 94974, 105239, 116218, 127935, 140414, 153679, 167754, 182663, 198430, 215079, 232634, 251119

Offset: 0

Author

Ilya Gutkovskiy, Feb 09 2016

Keywords

• easy
• sign

Comments

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.

Primes in this sequence: 199, 743, 15583, 105239, 435359, 620999, 770239, 853079, 1738423, 3511103, 7580119, 8737039, 10006063, ...

If a(n) is a positive prime then n is congruent to 0 or 4 (mod 6).

Links

• Ilya Gutkovskiy, Examples of the ordinary generating function for the values of cubic polynomialK
• Eric Weisstein's World of Mathematics, Cubic Polynomial
• Index entries for linear recurrences with constant coefficients, signature (4,-6,4,-1).

Crossrefs

Cf. A005408, A056578, A103532, A131464.

Programs

Magma

[4*n^3-3*n^2-2*n-1: n in [0..40]]; // Vincenzo Librandi, Feb 10 2016

Mathematica

Table[4 n^3 - 3 n^2 - 2 n - 1, {n, 0, 40}]
LinearRecurrence[{4, -6, 4, -1}, {-1, -2, 15, 74}, 41]
CoefficientList[Series[(-1 + 2 x + 17 x^2 + 6 x^3) / (1 - x)^4, {x, 0, 40}], x] (* Vincenzo Librandi, Feb 10 2016 *)

PARI

a(n)=4*n^3-3*n^2-2*n-1 \\ Charles R Greathouse IV, Jul 26 2016

Formula

G.f.: (-1 + 2*x + 17*x^2 + 6*x^3)/(1 - x)^4.

a(n) = A103532(n - 1) - A005408(n), n>0.

a(n) = 4*a(n - 1) - 6*a(n - 2) + 4*a(n - 3) - a(n - 4).

Sum_{n>=0} 1/a(n) = -1.407823506818026589265...

E.g.f.: exp(x)*(-1 - x + 9*x^2 + 4*x^3). - Stefano Spezia, Nov 17 2024