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a(n) = n*(n+1)*(n^2 - 3*n + 6)/4.
4

%I #20 Oct 21 2022 21:28:03

%S 0,2,6,18,50,120,252,476,828,1350,2090,3102,4446,6188,8400,11160,

%T 14552,18666,23598,29450,36330,44352,53636,64308,76500,90350,106002,

%U 123606,143318,165300,189720,216752,246576,279378,315350,354690,397602,444296

%N a(n) = n*(n+1)*(n^2 - 3*n + 6)/4.

%C a(n) = 1*2*3 + 2*3*4 + 3*4*5 +. . .+ (n-2)*(n-1)*n +(n-1)*n*1+ n*1*2, the sum of the cyclic product of terms taken three at a time, final term being n*1*2=2n.

%H Harry J. Smith, <a href="/A062026/b062026.txt">Table of n, a(n) for n = 0..1000</a>

%H <a href="/index/Rec#order_05">Index entries for linear recurrences with constant coefficients</a>, signature (5,-10,10,-5,1).

%F a(n) = 2 * A004255(n).

%F a(0)=0, a(1)=2, a(2)=6, a(3)=18, a(4)=50, a(n) = 5*a(n-1) - 10*a(n-2) + 10*a(n-3) - 5*a(n-4) + a(n-5). - _Harvey P. Dale_, Apr 22 2015

%F From _G. C. Greubel_, May 05 2022: (Start)

%F a(n) = 6*binomial(n+3, 4) - 12*binomial(n+2, 3) + 8*binomial(n+1, 2).

%F G.f.: 2*x*(1 - 2*x + 4*x^2)/(1-x)^5.

%F E.g.f.: (1/4)*x*(8 + 4*x + 4*x^2 + x^3)*exp(x). (End)

%e a(4) = 1*2*3 + 2*3*4 + 3*4*1 + 4*1*2 = 50.

%t Table[n(n+1)(n^2-3n+6)/4,{n,0,40}] (* or *) LinearRecurrence[{5,-10,10,-5,1},{0,2,6,18,50},40] (* _Harvey P. Dale_, Apr 22 2015 *)

%o (PARI) for (n=0, 100, write("b062026.txt", n, " ", n*(n+1)*(n^2 -3*n +6)/4) ) \\ _Harry J. Smith_, Jul 29 2009

%o (SageMath) [n*(n+1)*(n^2-3*n+6)/4 for n in (0..40)] # _G. C. Greubel_, May 05 2022

%Y Cf. A004255.

%K nonn,easy

%O 0,2

%A _Amarnath Murthy_, Jun 02 2001

%E More terms from Larry Reeves (larryr(AT)acm.org), Jun 06 2001