OFFSET
0,2
COMMENTS
(8*n^3 + n, 8*n^3 - n) solves the Diophantine equation 2*(X-Y)^3-(X+Y)=0.
(m*(2n)^k+n, m*(2n)^k-n) solves the Diophantine equation: 2m*(X-Y)^k-(X+Y)=0 with X>=Y,k>=2 and where m is a positive integer. Also ((m*n^k+n)/2, (m*n^k-n)/2) solves the Diophantine equation: m*(X-Y)^k-(X+Y)=0 with X>=Y,k>=2 where m is an odd number.
24*a(n) = (4n+1)^3 + (4n)^3 + (4n-1)^3. [Bruno Berselli, May 12 2014]
LINKS
Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (4,-6,4,-1).
FORMULA
G.f.: 3*x*(x+3)*(3*x+1)/(-1+x)^4. - R. J. Mathar, Nov 14 2007
a(n) = n*A081585(n). - Vincenzo Librandi, May 13 2014
MAPLE
MATHEMATICA
Table[8 n^3 + n, {n, 0, 35}]
CoefficientList[Series[3 x (x + 3) (3 x + 1)/(-1 + x)^4, {x, 0, 40}], x] (* Vincenzo Librandi, May 13 2014 *)
LinearRecurrence[{4, -6, 4, -1}, {0, 9, 66, 219}, 40] (* Harvey P. Dale, Feb 01 2023 *)
PROG
(Magma) [8*n^3 + n: n in [0..30]]; // Wesley Ivan Hurt, May 13 2014
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Mohamed Bouhamida, May 16 2006, Oct 02 2007
EXTENSIONS
Edited by Stefan Steinerberger, Jul 24 2007
STATUS
approved