A067728 a(n) = 2*n^2 + 8*n.

10, 24, 42, 64, 90, 120, 154, 192, 234, 280, 330, 384, 442, 504, 570, 640, 714, 792, 874, 960, 1050, 1144, 1242, 1344, 1450, 1560, 1674, 1792, 1914, 2040, 2170, 2304, 2442, 2584, 2730, 2880, 3034, 3192, 3354, 3520, 3690, 3864, 4042, 4224, 4410, 4600, 4794

Offset: 1

Author

Robert G. Wilson v, Feb 05 2002

Keywords

• nonn
• easy

Comments

Positive numbers k such that 8*(8 + k) is a perfect square.

Links

• Vincenzo Librandi, Table of n, a(n) for n = 1..1000
• Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Crossrefs

Cf. 7: A067727, 6: A067726, 5: A067724, 3: A067725.

Cf. A000217, A005563, A140091, A140681, A212331.

Programs

Magma

[2*n*(n+4): n in [1..50]] // Vincenzo Librandi, Jul 08 2012

Mathematica

Select[ Range[10000], IntegerQ[ Sqrt[ 8(8 + # )]] & ]
CoefficientList[Series[2*(5-3*x)/(1-x)^3,{x,0,50}],x] (* Vincenzo Librandi, Jul 08 2012 *)

PARI

a(n)=2*n*(n+4) \\ Charles R Greathouse IV, Dec 07 2011

Python

def a(n): return (2*n + 8)*n
print([a(n) for n in range(1, 48)]) # Michael S. Branicky, Oct 24 2021

Formula

a(n+1) = 2*n*n + 12*n + 10. - Frank Ellermann

a(n) = Sum_{k=0..n} Sum_{j=4..n} (j - k), n >= 4. - Zerinvary Lajos, May 11 2007

From Vincenzo Librandi, Jul 08 2012: (Start)

a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3).

G.f.: 2*x*(5-3*x)/(1-x)^3. (End)

From Amiram Eldar, Feb 25 2022: (Start)

Sum_{n>=1} 1/a(n) = 25/96.

Sum_{n>=1} (-1)^(n+1)/a(n) = 7/96. (End)

E.g.f.: 2*exp(x)*x*(5 + x). - Stefano Spezia, Oct 01 2023