A069140 a(n) = (4n-1)*4n*(4n+1).

0, 60, 504, 1716, 4080, 7980, 13800, 21924, 32736, 46620, 63960, 85140, 110544, 140556, 175560, 215940, 262080, 314364, 373176, 438900, 511920, 592620, 681384, 778596, 884640, 999900, 1124760, 1259604, 1404816, 1560780, 1727880, 1906500

Offset: 0

Author

Henry Bottomley, Apr 08 2002

Keywords

• easy
• nonn

Examples

			a(10) = 39*40*41 = 63960.
		

Links

• Harvey P. Dale, Table of n, a(n) for n = 0..1000
• Henry Bottomley, Medians and area bisectors of a triangle.
• S. Ramanujan, Notebook entry.
• Index entries for linear recurrences with constant coefficients, signature (4,-6,4,-1).

Crossrefs

Cf. A007531, A069072, A097321.

Programs

Mathematica

Table[64n^3-4n,{n,0,40}] (* or *) LinearRecurrence[{4,-6,4,-1},{0,60,504,1716},40] (* Harvey P. Dale, Dec 24 2012 *)

PARI

a(n)=(4*n-1)*4*n*(4*n+1) \\ Charles R Greathouse IV, Oct 07 2015

Formula

Sum_{i>0} 1/a(i) = log(2)*3/4 - 1/2 = 0.019860..., which is the ratio of the area of the deltoid envelope formed by area bisectors of a triangle to the area of the triangle.

a(n) = 64n^3 - 4n = A007531(4n) = A069072(2n-1).

a(n) = 4*a(n-1) - 6*a(n-2) + 4*a(n-3) - a(n-4). - Harvey P. Dale, Dec 24 2012

Sum_{n>=1} (-1)^(n+1)/a(n) = 1/2 - log(2)/4 + log(tan(Pi/8))/(2*sqrt(2)). Amiram Eldar, Mar 20 2022