A104188 - OEIS

%I #39 Nov 29 2024 17:35:10

%S 0,12,56,132,240,380,552,756,992,1260,1560,1892,2256,2652,3080,3540,

%T 4032,4556,5112,5700,6320,6972,7656,8372,9120,9900,10712,11556,12432,

%U 13340,14280,15252,16256,17292,18360,19460,20592,21756,22952,24180

%N a(n) = 4*n*(4*n - 1).

%C There is a ball-hating monster that lives in a box. You throw 4 numbered balls into the box. He throws 2 balls out. Repeat. Then a(n) gives the number of ordered possibilities the monster has to throw the balls back at each stage (2,1 is different from 1,2).

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

%F a(n) = (4*n)!/(4*n-2)! for n>0.

%F a(n) = 32*n + a(n-1) - 20 (with a(0)=0). - _Vincenzo Librandi_, Nov 13 2010

%F From _Colin Barker_, Jun 25 2012: (Start)

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

%F G.f.: 4*x*(3+5*x)/(1-x)^3. (End)

%F Sum_{n>=1} 1/a(n) = 3*log(2)/4 - Pi/8. - _Amiram Eldar_, Jan 03 2022

%F E.g.f.: 4*exp(x)*x*(3 + 4*x). - _Stefano Spezia_, Nov 29 2024

%e a(2) = (4*2)!/(4*2-2)! = 8!/6! = 8*7 = 56.

%p for n from 1 to 100 do printf(`%d,`, (4*n-4)*(4*n-5)) od: # _James A. Sellers_, Apr 10 2005

%o (PARI) a(n)=4*n*(4*n-1) \\ _Charles R Greathouse IV_, Jun 16 2017

%Y Cf. A004767, A008586.

%K nonn,easy

%O 0,2

%A Ruppi Rana (ruppi.rana(AT)gmail.com), Mar 12 2005

%E More terms from _James A. Sellers_, Apr 10 2005

%E Simpler definition from _Ralf Stephan_, May 20 2007