cp's OEIS Frontend

A137233 Number of n-digit even numbers.

%I A137233 #37 Jul 28 2025 19:54:24
%S A137233 5,45,450,4500,45000,450000,4500000,45000000,450000000,4500000000,
%T A137233 45000000000,450000000000,4500000000000,45000000000000,
%U A137233 450000000000000,4500000000000000,45000000000000000,450000000000000000,4500000000000000000,45000000000000000000,450000000000000000000
%N A137233 Number of n-digit even numbers.
%C A137233 From _Kival Ngaokrajang_, Oct 18 2013: (Start)
%C A137233 a(n) is also the total number of double rows identified numbers in n digit.
%C A137233 For example:
%C A137233   n = 1: 01 23 45 67 89 = 5 double rows;
%C A137233   n = 2: 1011 1213 1415 1617 1819...9899 = 45 double rows;
%C A137233   n = 3: 100101 102103 104105...998999 = 450 double rows;
%C A137233 The number of double rows is also A030656. (End)
%C A137233 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
%C A137233 a(n) is also the number of n-digit integers with an odd number of odd digits (A054684). - _Bernard Schott_, Nov 07 2022
%H A137233 Olympiade Mathématique Belge, <a href="http://omb.sbpm.be/modules/finale/article.php?storyid=86">OMB 2004, Finale Maxi, Question 2</a>.
%H A137233 <a href="/index/O#Olympiads">Index to sequences related to Olympiads</a>.
%H A137233 <a href="/index/Rec#order_01">Index entries for linear recurrences with constant coefficients</a>, signature (10).
%F A137233 a(n) = 9*10^(n-1)/2 if n > 1. - _R. J. Mathar_, May 23 2008
%F A137233 From _Elmo R. Oliveira_, Jul 23 2025: (Start)
%F A137233 G.f.: 5*x*(1-x)/(1-10*x).
%F A137233 E.g.f.: (-9 + 10*x + 9*exp(10*x))/20.
%F A137233 a(n) = 10*a(n-1) for n > 2.
%F A137233 a(n) = A052268(n)/2 for n >= 2. (End)
%e A137233 a(2) = 45 because there are 45 2-digit even numbers.
%o A137233 (Python)
%o A137233 def A137233(n): return 9*10**(n-1)+1>>1 # _Chai Wah Wu_, Nov 11 2022
%o A137233 (PARI) Vec(5*x*(1-x)/(1-10*x) + O(x^22)) \\ _Elmo R. Oliveira_, Jul 23 2025
%Y A137233 Cf. A000422, A002275, A002276, A011577, A014923, A014925, A016313, A019518, A037487, A053052, A057138, A090843, A097166, A099914, A099915.
%Y A137233 Cf. A030656, A052268, A054684, A356929.
%K A137233 easy,nonn,base
%O A137233 1,1
%A A137233 _Ctibor O. Zizka_, Mar 08 2008
%E A137233 Corrected and extended by _R. J. Mathar_, May 23 2008
%E A137233 More terms from _Elmo R. Oliveira_, Jul 23 2025