A135453 a(n) = 12*n^2.

0, 12, 48, 108, 192, 300, 432, 588, 768, 972, 1200, 1452, 1728, 2028, 2352, 2700, 3072, 3468, 3888, 4332, 4800, 5292, 5808, 6348, 6912, 7500, 8112, 8748, 9408, 10092, 10800, 11532, 12288, 13068, 13872, 14700, 15552, 16428, 17328, 18252, 19200, 20172, 21168, 22188

Offset: 0

Author

Ben Paul Thurston, Dec 14 2007

Keywords

• nonn
• easy

Comments

Areas of perfect 4:3 rectangles (for n > 0).

Sequence found by reading the line from 0, in the direction 0, 12, ..., in the square spiral whose vertices are the generalized octagonal numbers A001082. Semi-axis opposite to A069190 in the same spiral. - Omar E. Pol, Sep 16 2011

(x,y,z) = (-a(n), 1 + n*a(n), 1 - n*a(n)) are solutions of the Diophantine equation x^3 + 2*y^3 + 2*z^3 = 4. - XU Pingya, Apr 30 2022

Examples

			192 is on the list since 16*12 is a 4:3 rectangle with integer sides and an area of 192.
		

Links

• Ivan Panchenko, Table of n, a(n) for n = 0..10000
• John Elias, Illustration: Even Ordered Star Perimeters.
• Leo Tavares, Illustration.
• Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Crossrefs

Cf. A000290, A001105, A016742, A033428, A033581, A069190, A086729.

Cf. A008594, A195143.

Programs

GAP

List([0..100],n->12*n^2); # Muniru A Asiru, Jan 29 2018

Maple

seq(12*h^2,n=0..100); # Muniru A Asiru, Jan 29 2018

Mathematica

Table[12*n^2, {n, 0, 60}] (* Stefan Steinerberger, Dec 17 2007 *)
LinearRecurrence[{3,-3,1},{0,12,48},50] (* Harvey P. Dale, Jan 19 2020 *)

PARI

a(n)=12*n^2 \\ Charles R Greathouse IV, Jun 17 2017

Formula

a(n) = 12*A000290(n) = 6*A001105(n) = 4*A033428(n) = 3*A016742(n) = 2*A033581(n). - Omar E. Pol, Dec 13 2008

From Amiram Eldar, Feb 03 2021: (Start)

Sum_{n>=1} 1/a(n) = Pi^2/72 (A086729).

Sum_{n>=1} (-1)^(n+1)/a(n) = Pi^2/144.

Product_{n>=1} (1 + 1/a(n)) = 2*sqrt(3)*sinh(Pi/(2*sqrt(3)))/Pi.

Product_{n>=1} (1 - 1/a(n)) = 2*sqrt(3)*sin(Pi/(2*sqrt(3)))/Pi. (End)

From Elmo R. Oliveira, Nov 30 2024: (Start)

G.f.: 12*x*(1 + x)/(1-x)^3.

E.g.f.: 12*x*(1 + x)*exp(x).

a(n) = n*A008594(n) = A195143(2*n).

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

Extensions

More terms from Stefan Steinerberger, Dec 17 2007

Minor edits from Omar E. Pol, Dec 15 2008