A081045 10th binomial transform of (1,9,0,0,0,0,0,...).

1, 19, 280, 3700, 46000, 550000, 6400000, 73000000, 820000000, 9100000000, 100000000000, 1090000000000, 11800000000000, 127000000000000, 1360000000000000, 14500000000000000, 154000000000000000, 1630000000000000000, 17200000000000000000, 181000000000000000000

Offset

0,2

Comments

From Bernard Schott, Nov 12 2022: (Start)

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.

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}.

First difference of A053541. (End)

Links

Vincenzo Librandi, Table of n, a(n) for n = 0..300

Index entries for linear recurrences with constant coefficients, signature (20,-100).

Formula

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

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

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

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

a(n) = A053541(n+1) - A053541(n), for n >= 1. - Bernard Schott, Nov 12 2022

E.g.f.: exp(10*x)*(1 + 9*x). - Stefano Spezia, Jan 31 2025

Mathematica

CoefficientList[Series[(1 - x)/(1 - 10 x)^2, {x, 0, 30}], x] (* Vincenzo Librandi, Aug 06 2013 *)
LinearRecurrence[{20, -100}, {1, 19}, 20] (* Harvey P. Dale, Dec 28 2023 *)

Prog

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

Crossrefs

Cf. A053541, A081044, A081043, A212704.

Sequence in context: A322628 A322053 A328916 * A155017 A199245 A152591

Adjacent sequences: A081042 A081043 A081044 * A081046 A081047 A081048

Keyword

easy,base,nonn,changed

Author

Paul Barry, Mar 04 2003

Status

approved