A294315 a(n) = 3*n^3 + n^2.

0, 4, 28, 90, 208, 400, 684, 1078, 1600, 2268, 3100, 4114, 5328, 6760, 8428, 10350, 12544, 15028, 17820, 20938, 24400, 28224, 32428, 37030, 42048, 47500, 53404, 59778, 66640, 74008, 81900, 90334, 99328, 108900, 119068, 129850, 141264, 153328, 166060, 179478

Offset

0,2

Comments

All terms are even.

Links

Muniru A Asiru, Table of n, a(n) for n = 0..10000 (first 1000 terms from Colin Barker)

Index entries for linear recurrences with constant coefficients, signature (4,-6,4,-1).

Formula

a(n) = 3*n^3 + n^2.

a(n) = A117642(n) + A000290(n).

a(n) = 2*A036659(n).

From Colin Barker, Dec 11 2017: (Start)

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

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

(End)

From Amiram Eldar, Jan 10 2023: (Start)

Sum_{n>=1} 1/a(n) = Pi^2/6 + sqrt(3)*Pi/2 + 9*log(3)/2 - 9.

Sum_{n>=1} (-1)^(n+1)/a(n) = Pi^2/12 - sqrt(3)*Pi - 6*log(2) + 9. (End)

Example

a(3)=90 because 3*3^3 + 3^2 = 3*27 + 9 = 90.

Mathematica

Array[3 #^3 + #^2 &, 40, 0] (* or *)
LinearRecurrence[{4, -6, 4, -1}, {0, 4, 28, 90}, 40] (* or *)
CoefficientList[Series[2 x (2 + 6 x + x^2)/(1 - x)^4, {x, 0, 39}], x] (* Michael De Vlieger, Dec 12 2017 *)

Prog

(PARI) a(n) = 3*n^3 + n^2;
(PARI) concat(0, Vec(2*x*(2 + 6*x + x^2) / (1 - x)^4 + O(x^40))) \\ Colin Barker, Dec 11 2017
(GAP) A294315:=List([0..10^4], n -> 3 *n^3 + n^2 ); # Muniru A Asiru, Dec 11 2017

Crossrefs

Cf. A117642, A000290, A036659.

Sequence in context: A005634 A183485 A183437 * A263239 A296015 A187452

Adjacent sequences: A294312 A294313 A294314 * A294316 A294317 A294318

Keyword

nonn,easy

Author

Jason Morgan, Oct 28 2017

Status

approved