A117727 Partial sums of A051109.
1, 3, 8, 18, 38, 88, 188, 388, 888, 1888, 3888, 8888, 18888, 38888, 88888, 188888, 388888, 888888, 1888888, 3888888, 8888888, 18888888, 38888888, 88888888, 188888888, 388888888, 888888888, 1888888888, 3888888888, 8888888888
Offset: 0
Links
- G. C. Greubel, Table of n, a(n) for n = 0..1000
- Index entries for linear recurrences with constant coefficients, signature (1,0,10,-10).
Programs
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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
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Mathematica
LinearRecurrence[{1,0,10,-10}, {1,3,8,18}, 41] (* G. C. Greubel, Jul 23 2023 *) -
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
Formula
a(n) = Sum_{j=0..n} A051109(j).
From G. C. Greubel, Jul 23 2023: (Start)
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).
a(n) = a(n-1) + 10*a(n-3) - 10*a(n-4).
G.f.: (1 + 2*x + 5*x^2)/((1 - x)*(1 - 10*x^3)). (End)