A017139 a(n) = (8*n + 6)^3.

216, 2744, 10648, 27000, 54872, 97336, 157464, 238328, 343000, 474552, 636056, 830584, 1061208, 1331000, 1643032, 2000376, 2406104, 2863288, 3375000, 3944312, 4574296, 5268024, 6028568, 6859000, 7762392, 8741816, 9800344, 10941048, 12167000, 13481272, 14886936

Offset: 0

Author

N. J. A. Sloane

Keywords

• nonn
• easy

Comments

4*n + 3 = (8*n + 6) / 2 is never a square, as 3 is not a quadratic residue modulo 4. Using this, we can show that each term has an even square part and an even squarefree part, neither part being a power of 2. (Less than 2% of integers have this property - see A339245.) - Peter Munn, Dec 14 2020

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..10000
• Eric Weisstein's World of Mathematics, Quadratic Residue.
• Index entries for linear recurrences with constant coefficients, signature (4,-6,4,-1).

Crossrefs

A000578, A016839, A017137 are used in a formula defining this sequence.

Subsequence of A339245.

Cf. A002117, A017138, A017140.

Programs

Magma

[(8*n+6)^3: n in [0..35]]; // Vincenzo Librandi, Jul 22 2011

Mathematica

Table[(8*n+6)^3,{n,0,5!}] (* Vladimir Joseph Stephan Orlovsky, Mar 17 2010 *)
LinearRecurrence[{4,-6,4,-1},{216,2744,10648,27000},30] (* Harvey P. Dale, Dec 11 2012 *)

Formula

From R. J. Mathar, Mar 22 2010: (Start)

G.f.: 8*(27 + 235*x + 121*x^2 + x^3)/(x-1)^4.

a(n) = 8*A016839(n). (End)

a(0)=216, a(1)=2744, a(2)=10648, a(3)=27000, a(n) = 4*a(n-1) - 6*a(n-2) + 4*a(n-3) - a(n-4). - Harvey P. Dale, Dec 11 2012

a(n) = A017137(n)^3 = A000578(A017137(n)). - Peter Munn, Dec 20 2020

Sum_{n>=0} 1/a(n) = 7*zeta(3)/128 - Pi^2/512. - Amiram Eldar, Apr 26 2023

Extensions

More terms from Vladimir Joseph Stephan Orlovsky, Mar 17 2010