A117727 - OEIS

%I #9 Jul 23 2023 20:21:41

%S 1,3,8,18,38,88,188,388,888,1888,3888,8888,18888,38888,88888,188888,

%T 388888,888888,1888888,3888888,8888888,18888888,38888888,88888888,

%U 188888888,388888888,888888888,1888888888,3888888888,8888888888

%N Partial sums of A051109.

%H G. C. Greubel, <a href="/A117727/b117727.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 (1,0,10,-10).

%F a(n) = Sum_{j=0..n} A051109(j).

%F From _G. C. Greubel_, Jul 23 2023: (Start)

%F a(n) = (1/9)*( -8 + 17*b(n) + 35*b(n-1) + 80*b(n-2) ), where b(n) = 10^floor(n/3)*floor((n-1 mod 3)/2).

%F a(n) = a(n-1) + 10*a(n-3) - 10*a(n-4).

%F G.f.: (1 + 2*x + 5*x^2)/((1 - x)*(1 - 10*x^3)). (End)

%t LinearRecurrence[{1,0,10,-10}, {1,3,8,18}, 41] (* _G. C. Greubel_, Jul 23 2023 *)

%o (Magma) I:=[1,3,8,18]; [n le 4 select I[n] else Self(n-1) +10*Self(n-3) -10*Self(n-4): n in [1..40]]; // _G. C. Greubel_, Jul 23 2023

%o (SageMath) [sum((1 + (j%3)^2)*10^(j//3) for j in range(n+1)) for n in range(41)] # _G. C. Greubel_, Jul 23 2023

%Y Cf. A051109, A117717.

%K nonn,easy

%O 0,2

%A _N. J. A. Sloane_, Apr 14 2006