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10th binomial transform of (1,9,0,0,0,0,0,...).
7

%I #43 Dec 28 2023 18:51:36

%S 1,19,280,3700,46000,550000,6400000,73000000,820000000,9100000000,

%T 100000000000,1090000000000,11800000000000,127000000000000,

%U 1360000000000000,14500000000000000,154000000000000000

%N 10th binomial transform of (1,9,0,0,0,0,0,...).

%C From _Bernard Schott_, Nov 12 2022: (Start)

%C For n >= 1, a(n-1) is the number of digits 1 (or any nonzero digit) that are necessary to write all the n-digit integers, while the corresponding number of digits 0 to write all these n-digit integers is A212704(n-1) for n >=2.

%C E.g.: a(2-1) = 19 since 19 digits 2's are required to write integers with a digit 2 from 10 up to 99: {12, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 32, 42, 52, 62, 72, 82, 92}.

%C First difference of A053541. (End)

%H Vincenzo Librandi, <a href="/A081045/b081045.txt">Table of n, a(n) for n = 0..300</a>

%H <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (20,-100).

%F a(n) = 20*a(n-1) - 100*a(n-2); a(0)=1, a(1)=19.

%F a(0)=1; for n>= 1, a(n) = (9*n+10)*10^(n-1) = 10^(n-1)*A017173(n+1).

%F a(n) = Sum_{k=0..n} (k+1)*9^k*binomial(n, k).

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

%F a(n) = A053541(n+1) - A053541(n), for n >= 1. - _Bernard Schott_, Nov 12 2022

%t CoefficientList[Series[(1 - x)/(1 - 10 x)^2, {x, 0, 30}], x] (* _Vincenzo Librandi_, Aug 06 2013 *)

%t LinearRecurrence[{20,-100},{1,19},20] (* _Harvey P. Dale_, Dec 28 2023 *)

%o (Magma) [(9*n+10)*10^(n-1): n in [0..25]]; // _Vincenzo Librandi_, Aug 06 2013

%Y Cf. A053541, A081044, A081043, A212704.

%K easy,base,nonn

%O 0,2

%A _Paul Barry_, Mar 04 2003