A280304 a(n) = 3*n*(n^2 + 3*n + 4).

0, 24, 84, 198, 384, 660, 1044, 1554, 2208, 3024, 4020, 5214, 6624, 8268, 10164, 12330, 14784, 17544, 20628, 24054, 27840, 32004, 36564, 41538, 46944, 52800, 59124, 65934, 73248, 81084, 89460, 98394, 107904, 118008, 128724, 140070, 152064, 164724, 178068, 192114, 206880, 222384, 238644, 255678, 273504, 292140, 311604, 331914, 353088, 375144, 398100

Offset: 0

Author

Luce ETIENNE, Dec 31 2016

Keywords

• nonn
• easy

Comments

Numbers of unit triangles in a certain structure obtained from A006003.

Examples

			a(0) = 6*(1-1) = 0, a(1) = 6*(5-1) = 24, a(2) = 6*(15-1) = 84, a(3) = 6*(34-1) = 198, a(4) = 6*(65-1) = 384.
		

Links

• Colin Barker, Table of n, a(n) for n = 0..1000
• Luce ETIENNE, Illustration of initial terms
• Index entries for linear recurrences with constant coefficients, signature (4,-6,4,-1).

Crossrefs

Cf. A003215, A005448, A006003, A033428, A213389, A269064.

Programs

Magma

[3*n*(n^2 + 3*n + 4) : n in [0..60]]; // Wesley Ivan Hurt, Dec 31 2016

Maple

A280304:=n->3*n*(n^2 + 3*n + 4): seq(A280304(n), n=0..60); # Wesley Ivan Hurt, Dec 31 2016

Mathematica

Table[3 n (n^2 + 3 n + 4), {n, 0, 50}] (* or *)
CoefficientList[Series[6 x (x^2 - 2 x + 4)/(1 - x)^4, {x, 0, 50}], x] (* Michael De Vlieger, Dec 31 2016 *)
LinearRecurrence[{4,-6,4,-1},{0,24,84,198},60] (* Harvey P. Dale, Feb 08 2023 *)

PARI

concat(0, Vec(6*x*(x^2-2*x+4) / (1-x)^4 + O(x^30))) \\ Colin Barker, Dec 31 2016

Formula

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

a(n) = 6*(A006003(n+1)-1).

a(n) = 4*a(n-1) - 6*a(n-2) + 4*a(n-3) - a(n-4) for n > 3. - Colin Barker, Dec 31 2016