A137233 Number of n-digit even numbers.

5, 45, 450, 4500, 45000, 450000, 4500000, 45000000, 450000000, 4500000000, 45000000000, 450000000000, 4500000000000, 45000000000000, 450000000000000, 4500000000000000, 45000000000000000, 450000000000000000, 4500000000000000000, 45000000000000000000, 450000000000000000000

Offset: 1

Author

Ctibor O. Zizka, Mar 08 2008

Keywords

• easy
• nonn
• base

Comments

From Kival Ngaokrajang, Oct 18 2013: (Start)

a(n) is also the total number of double rows identified numbers in n digit.

For example:

n = 1: 01 23 45 67 89 = 5 double rows;

n = 2: 1011 1213 1415 1617 1819...9899 = 45 double rows;

n = 3: 100101 102103 104105...998999 = 450 double rows;

The number of double rows is also A030656. (End)

a(n) is also the number of n-digit integers with an even number of even digits (A356929); a(5) = 45000 is the answer to the question 2 of the Olympiade Mathématique Belge in 2004 (link). - Bernard Schott, Sep 06 2022

a(n) is also the number of n-digit integers with an odd number of odd digits (A054684). - Bernard Schott, Nov 07 2022

Examples

			a(2) = 45 because there are 45 2-digit even numbers.
		

Links

• Olympiade Mathématique Belge, OMB 2004, Finale Maxi, Question 2.
• Index to sequences related to Olympiads.
• Index entries for linear recurrences with constant coefficients, signature (10).

Crossrefs

Cf. A000422, A002275, A002276, A011577, A014923, A014925, A016313, A019518, A037487, A053052, A057138, A090843, A097166, A099914, A099915.

Cf. A030656, A052268, A054684, A356929.

Programs

PARI

Vec(5*x*(1-x)/(1-10*x) + O(x^22)) \\ Elmo R. Oliveira, Jul 23 2025

Python

def A137233(n): return 9*10**(n-1)+1>>1 # Chai Wah Wu, Nov 11 2022

Formula

a(n) = 9*10^(n-1)/2 if n > 1. - R. J. Mathar, May 23 2008

From Elmo R. Oliveira, Jul 23 2025: (Start)

G.f.: 5*x*(1-x)/(1-10*x).

E.g.f.: (-9 + 10*x + 9*exp(10*x))/20.

a(n) = 10*a(n-1) for n > 2.

a(n) = A052268(n)/2 for n >= 2. (End)

Extensions

Corrected and extended by R. J. Mathar, May 23 2008

More terms from Elmo R. Oliveira, Jul 23 2025