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a(n) = 3*(10^n) + (n^2 + 1)*(10^n - 1)/9.
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%I #13 Jan 05 2022 17:34:55

%S 3,32,355,4110,48887,588886,7111107,85555550,1022222215,12111111102,

%T 142222222211,1655555555542,19111111111095,218888888888870,

%U 2488888888888867,28111111111111086,315555555555555527,3522222222222222190,39111111111111111075,432222222222222222182

%N a(n) = 3*(10^n) + (n^2 + 1)*(10^n - 1)/9.

%C Sequence generalized: a(n) = a(0)*(B^n) + F(n)* ((B^n)-1)/(B-1); a(0), B integers, F(n) arithmetic function.

%C Examples:

%C a(0) = 1, B = 10, F(n) = 1 gives A002275, F(n) = 2 gives A090843, F(n) = 3 gives A097166, F(n) = 4 gives A099914, F(n) = 5 gives A099915.

%C a(0) = 1, B = 2, F(n) = 1 gives A000225, F(n) = 2 gives A033484, F(n) = 3 gives A036563, F(n) = 4 gives A048487, F(n) = 5 gives A048488, F(n) = 6 gives A048489.

%C a(0) = 1, B = 3, F(n) = 1 gives A003462, F(n) = 2 gives A048473, F(n) = 3 gives A134931, F(n) = 4 gives A058481, F(n) = 5 gives A116952.

%C a(0) = 1, B = 4, F(n) = 1 gives A002450, F(n) = 2 gives A020989, F(n) = 3 gives A083420, F(n) = 4 gives A083597, F(n) = 5 gives A083584.

%C a(0) = 1, B = 5, F(n) = 1 gives A003463, F(n) = 2 gives A057651, F(n) = 3 gives A117617, F(n) = 4 gives A081655.

%C a(0) = 2, B = 10, F(n) = 1 gives A037559, F(n) = 2 gives A002276.

%H G. C. Greubel, <a href="/A137215/b137215.txt">Table of n, a(n) for n = 0..985</a>

%H <a href="/index/Rec#order_06">Index entries for linear recurrences with constant coefficients</a>, signature (33,-393,1991,-3930,3300,-1000).

%F a(n) = 3*(10^n) + (n^2 + 1)*(10^n - 1)/9.

%F O.g.f.: (3 - 67*x + 478*x^2 - 1002*x^3 + 850*x^4 - 100*x^5)/((1-x)^3 * (1-10*x)^3). - _R. J. Mathar_, Mar 16 2008

%e a(3) = 3*10^3 + (3*3 + 1)*(10^3 - 1)/9 = 4110.

%t Table[3*10^n +(n^2 +1)*(10^n -1)/9, {n,0,30}] (* _G. C. Greubel_, Jan 05 2022 *)

%o (PARI) a(n) = 3*(10^n) + (n*n+1)*((10^n)-1)/9; \\ _Jinyuan Wang_, Feb 27 2020

%o (Sage) [3*10^n +(1+n^2)*(10^n -1)/9 for n in (0..30)] # _G. C. Greubel_, Jan 05 2022

%Y Cf. A002275, A011557.

%K nonn,easy

%O 0,1

%A _Ctibor O. Zizka_, Mar 06 2008

%E More terms from _R. J. Mathar_, Mar 16 2008

%E More terms from _Jinyuan Wang_, Feb 27 2020