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Partial sums of A001580.
1

%I #9 Sep 23 2019 16:52:24

%S 1,4,12,29,61,118,218,395,715,1308,2432,4601,8841,17202,33782,66775,

%T 132567,263928,526396,1051045,2100021,4197614,8392402,16781539,

%U 33559331,67114388,134223928,268442385,536878625,1073750378,2147493102,4294977711,8589946031

%N Partial sums of A001580.

%C A001580 2^n+n^2 -> 1,3,8,17,32,57,100,177,320,593,1124,..

%H Colin Barker, <a href="/A174121/b174121.txt">Table of n, a(n) for n = 1..1000</a>

%H <a href="/index/Rec#order_05">Index entries for linear recurrences with constant coefficients</a>, signature (6,-14,16,-9,2).

%F From _Colin Barker_, Feb 26 2016: (Start)

%F a(n) = (n-2)*(2*n^2+n+3)/6+2^n.

%F a(n) = 6*a(n-1)-14*a(n-2)+16*a(n-3)-9*a(n-4)+2*a(n-5) for n>5.

%F G.f.: x*(1-2*x+2*x^2-3*x^3) / ((1-x)^4*(1-2*x)).

%F (End)

%t f[n_]:=Sum[2^i+i^2,{i,0,n}];Table[f[n],{n,0,5!}]

%t Accumulate[Table[2^n+n^2,{n,0,50}]] (* or *) LinearRecurrence[{6,-14,16,-9,2},{1,4,12,29,61},50] (* _Harvey P. Dale_, Sep 23 2019 *)

%o (PARI) Vec(x*(1-2*x+2*x^2-3*x^3)/((1-x)^4*(1-2*x)) + O(x^40)) \\ _Colin Barker_, Feb 26 2016

%Y Cf. A174120.

%K nonn,easy

%O 1,2

%A _Vladimir Joseph Stephan Orlovsky_, Mar 08 2010