A067728 - OEIS

%I #52 Oct 01 2023 08:59:14

%S 10,24,42,64,90,120,154,192,234,280,330,384,442,504,570,640,714,792,

%T 874,960,1050,1144,1242,1344,1450,1560,1674,1792,1914,2040,2170,2304,

%U 2442,2584,2730,2880,3034,3192,3354,3520,3690,3864,4042,4224,4410,4600,4794

%N a(n) = 2*n^2 + 8*n.

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

%H Vincenzo Librandi, <a href="/A067728/b067728.txt">Table of n, a(n) for n = 1..1000</a>

%H <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (3,-3,1).

%F a(n+1) = 2*n*n + 12*n + 10. - _Frank Ellermann_

%F a(n) = Sum_{k=0..n} Sum_{j=4..n} (j - k), n >= 4. - _Zerinvary Lajos_, May 11 2007

%F From _Vincenzo Librandi_, Jul 08 2012: (Start)

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

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

%F From _Amiram Eldar_, Feb 25 2022: (Start)

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

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

%F E.g.f.: 2*exp(x)*x*(5 + x). - _Stefano Spezia_, Oct 01 2023

%t Select[ Range[10000], IntegerQ[ Sqrt[ 8(8 + # )]] & ]

%t CoefficientList[Series[2*(5-3*x)/(1-x)^3,{x,0,50}],x] (* _Vincenzo Librandi_, Jul 08 2012 *)

%o (PARI) a(n)=2*n*(n+4) \\ _Charles R Greathouse IV_, Dec 07 2011

%o (Magma) [2*n*(n+4): n in [1..50]] // _Vincenzo Librandi_, Jul 08 2012

%o (Python)

%o def a(n): return (2*n + 8)*n

%o print([a(n) for n in range(1, 48)]) # _Michael S. Branicky_, Oct 24 2021

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

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

%K nonn,easy

%O 1,1

%A _Robert G. Wilson v_, Feb 05 2002