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a(n) = 10*a(n-1)+n^2, a(0)=0.
1

%I #19 May 25 2024 15:53:57

%S 0,1,14,149,1506,15085,150886,1508909,15089154,150891621,1508916310,

%T 15089163221,150891632354,1508916323709,15089163237286,

%U 150891632373085,1508916323731106,15089163237311349,150891632373113814

%N a(n) = 10*a(n-1)+n^2, a(0)=0.

%C a(n)/10^n converges to 110/729=0.15089163237311...

%H Vincenzo Librandi, <a href="/A032343/b032343.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 (13,-33,31,-10).

%F a(n) =(10^n-1)*(110/729)-n^2/9-n*(20/81).

%F G.f.: (x^2+x)/((x-1)^3*(10*x-1)). - _Harvey P. Dale_, Mar 20 2011

%t CoefficientList[Series[(x^2+x)/((x-1)^3 (10x-1)),{x,0,30}],x] (* _Harvey P. Dale_, Mar 20 2011 *)

%t RecurrenceTable[{a[0] == 0, a[n] == 10*a[n-1] + n^2}, a, {n, 100}] (* _Vincenzo Librandi_, Mar 10 2013 *)

%t LinearRecurrence[{13,-33,31,-10},{0,1,14,149},20] (* _Harvey P. Dale_, May 25 2024 *)

%o (Magma) [0] cat [n le 1 select n else 10*Self(n-1)+n^2: n in [1..20]]; // _Vincenzo Librandi_, Mar 10 2013

%Y Cf. A014824, A019521.

%K nonn,easy

%O 0,3

%A _Henry Bottomley_, Jul 04 2000