A144965 a(n) = 4*n*(4*n^2 + 1).

0, 20, 136, 444, 1040, 2020, 3480, 5516, 8224, 11700, 16040, 21340, 27696, 35204, 43960, 54060, 65600, 78676, 93384, 109820, 128080, 148260, 170456, 194764, 221280, 250100, 281320, 315036, 351344, 390340, 432120, 476780, 524416, 575124, 629000, 686140, 746640

Offset: 0

Author

Luc Comeau-Montasse, Sep 27 2008

Keywords

• easy
• nonn

Comments

(a(n))^2 + (n*a(n)+1)^2 is always a perfect square.

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..1000
• Luc Comeau-Montasse, Des mesures entières pour des triangles rectangles, Géométrie et nombre (French blog), September 2008.
• Index entries for linear recurrences with constant coefficients, signature (4,-6,4,-1).

Crossrefs

Cf. A008586, A053755, A317297.

Programs

Magma

I:=[0, 20, 136, 444]; [n le 4 select I[n] else 4*Self(n-1)-6*Self(n-2)+4*Self(n-3)-Self(n-4): n in [1..50]]; // Vincenzo Librandi, Jun 30 2012

Mathematica

CoefficientList[Series[4*x*(5+14*x+5*x^2)/(1-x)^4,{x,0,40}],x] (* Vincenzo Librandi, Jun 30 2012 *)
LinearRecurrence[{4,-6,4,-1},{0,20,136,444},50] (* Harvey P. Dale, Aug 07 2022 *)

Formula

G.f.: 4*x*(5+14*x+5*x^2)/(1-x)^4. - Colin Barker, May 24 2012

a(n) = 4*a(n-1) - 6*a(n-2) + 4*a(n-3) - a(n-4). - Vincenzo Librandi, Jun 30 2012

From Elmo R. Oliveira, Aug 07 2025: (Start)

E.g.f.: 4*x*(5 + 2*x)*(1 + 2*x)*exp(x).

a(n) = 4*A317297(n+1) = A008586(n)*A053755(n). (End)