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%I #31 Sep 27 2023 16:43:18
%S 0,9,50,147,324,605,1014,1575,2312,3249,4410,5819,7500,9477,11774,
%T 14415,17424,20825,24642,28899,33620,38829,44550,50807,57624,65025,
%U 73034,81675,90972,100949,111630,123039,135200,148137,161874
%N a(n) = n*(2*n+1)^2.
%H Vincenzo Librandi, <a href="/A084367/b084367.txt">Table of n, a(n) for n = 0..1000</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) = n*( n*(2*n+1)+1 + n*(2*n+1)+2 + ... + n*(2*n+1)+2*n ).
%F a(n) = n*A016754(n); n*a(n) = A014105(n)^2.
%F G.f.: x*(9+14*x+x^2)/(1-x)^4. - _Colin Barker_, Jun 30 2012
%F a(n) = 4*a(n-1) -6*a(n-2) +4*a(n-3) -a(n-4). - _Vincenzo Librandi_, Jul 04 2012
%F Sum_{n>=1} 1/a(n) = 4 - 2*log(2) - Pi^2/4. - _Amiram Eldar_, Jul 21 2020
%F Sum_{n>=1} (-1)^(n+1)/a(n) = Pi/2 + log(2) + 2*G - 4, where G is Catalan's constant (A006752). - _Amiram Eldar_, Feb 08 2022
%F E.g.f.: exp(x)*x*(9 + 16*x + 4*x^2). - _Stefano Spezia_, Sep 27 2023
%e a(3) = 147 since 147 = 3*7^2.
%t CoefficientList[Series[x*(9+14*x+x^2)/(1-x)^4,{x,0,50}],x] (* _Vincenzo Librandi_, Jul 04 2012 *)
%o (Magma) I:=[0, 9, 50, 147]; [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..40]]; // _Vincenzo Librandi_, Jul 04 2012
%Y Cf. A006752, A014105, A016754.
%K easy,nonn
%O 0,2
%A _Charlie Marion_, Jun 22 2003
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