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%I #33 Sep 08 2022 08:46:01
%S 0,2,10,29,64,120,202,315,464,654,890,1177,1520,1924,2394,2935,3552,
%T 4250,5034,5909,6880,7952,9130,10419,11824,13350,15002,16785,18704,
%U 20764,22970,25327,27840,30514,33354,36365,39552,42920,46474,50219
%N a(n) = n*(5n^2 + 3n + 4) / 6.
%H Vincenzo Librandi, <a href="/A203551/b203551.txt">Table of n, a(n) for n = 0..10000</a>
%H <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (4,-6,4,-1).
%F a(n) = Sum_{k = 1..n} A(-k, k-n-1) where A(i, j) = i^2 + i*j + j^2 + i + j + 1.
%F G.f.: x * (2 + 2*x + x^2) / (1 - x)^4.
%F a(n) = -A203552(-n) for all n in Z.
%F a(n)-a(n-1) = A192136(n). - _Bruno Berselli_, Jan 03 2012
%F E.g.f.: x*(5*x^2 + 18*x + 12)*exp(x)/6. - _G. C. Greubel_, Aug 12 2018
%e G.f. = 2*x + 10*x^2 + 29*x^3 + 64*x^4 + 120*x^5 + 202*x^6 + 315*x^7 + 464*x^8 + ...
%t LinearRecurrence[{4,-6,4,-1},{0,2,10,29},40] (* _Vincenzo Librandi_, Jan 07 2012 *)
%t Table[n (5n^2+3n+4)/6,{n,0,40}] (* _Harvey P. Dale_, Mar 24 2022 *)
%o (PARI) {a(n) = n * (5*n^2 + 3*n + 4) / 6};
%o (Magma) I:=[0, 2, 10, 29]; [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_, Jan 07 2012
%Y Cf. A203552.
%K nonn,easy
%O 0,2
%A _Michael Somos_, Jan 02 2012
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