OFFSET
0,2
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
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).
FORMULA
a(n) = A000290(n)*12 = A001105(n)*6 = A033428(n)*4 = A016742(n)*3 = A033581(n)*2. - 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)
EXAMPLE
192 is on the list since 16*12 is a 4:3 rectangle with integer sides and an area of 192.
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 *)
PROG
(PARI) a(n)=12*n^2 \\ Charles R Greathouse IV, Jun 17 2017
(GAP) List([0..100], n->12*n^2); # Muniru A Asiru, Jan 29 2018
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Ben Paul Thurston, Dec 14 2007
EXTENSIONS
More terms from Stefan Steinerberger, Dec 17 2007
Minor edits from Omar E. Pol, Dec 15 2008
STATUS
approved