A358620 Number of nonzero digits needed to write all nonnegative n-digit integers.

9, 171, 2520, 33300, 414000, 4950000, 57600000, 657000000, 7380000000, 81900000000, 900000000000, 9810000000000, 106200000000000, 1143000000000000, 12240000000000000, 130500000000000000, 1386000000000000000, 14670000000000000000, 154800000000000000000

Offset

1,1

Links

Table of n, a(n) for n=1..19.

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

Formula

a(n) = 9*(9*n+1)*10^(n-2).

a(n) = 20*a(n-1) - 100*a(n-2); a(1)=9, a(2)=171.

a(n) = 9*A081045(n-1).

a(n) = A113119(n) - A212704(n-1), for n >= 2.

From Elmo R. Oliveira, Nov 19 2025: (Start)

G.f.: 9*x*(1-x)/(10*x-1)^2.

E.g.f.: 9*(exp(10*x)*(90*x+1) - 1)/100. (End)

Example

a(1) = 9 because there are 9 one-digit numbers that are > 0.
a(2) = 171 because there are 90 two-digit numbers, so 90*2 = 180 digits are needed to write these integers, nine of these integers end with 0, and 180-9 = 171.

Maple

seq((9*(9*n+1))*10^(n-2), n = 1 .. 20);

Mathematica

a[n_] := 9*(9*n + 1)*10^(n - 2); Array[a, 20] (* Amiram Eldar, Nov 23 2022 *)

Prog

(PARI) a(n)=(81*n+9)*10^(n-2) \\ Charles R Greathouse IV, Nov 29 2022
(Python)
def A358620(n): return 9 if n == 1 else 9*(9*n+1)*10**(n-2) # Chai Wah Wu, Nov 29 2022

Crossrefs

Cf. A081045, A113119, A212704.

Sequence in context: A055407 A083155 A229127 * A358959 A132897 A238466

Adjacent sequences: A358617 A358618 A358619 * A358621 A358622 A358623

Keyword

nonn,base,easy

Author

Bernard Schott, Nov 23 2022

Status

approved