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) = 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
KEYWORD
nonn,easy
AUTHOR
Jason Morgan, Oct 28 2017
STATUS
approved