A030141 Numbers in which parity of the decimal digits alternates.

0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 12, 14, 16, 18, 21, 23, 25, 27, 29, 30, 32, 34, 36, 38, 41, 43, 45, 47, 49, 50, 52, 54, 56, 58, 61, 63, 65, 67, 69, 70, 72, 74, 76, 78, 81, 83, 85, 87, 89, 90, 92, 94, 96, 98, 101, 103, 105, 107, 109, 121, 123, 125, 127, 129

Offset: 1

Author

Patrick De Geest

Keywords

• nonn
• base
• easy

Comments

An alternating integer is a positive integer for which, in base-10, the parity of its digits alternates.

The number of terms < 10^n (n>=0): 1, 10, 55, 280, 1405, 7030, 35155, ..., . - Robert G. Wilson v, Apr 01 2011

The number of terms between 10^n and 10^(n+1) is 9 * 5^n for n>=0. For n>=0, number of terms < 10^n is 1 + 9 * (5^n-1)/4. - Franklin T. Adams-Watters, Apr 01 2011

A228710(a(n)) = 1. - Reinhard Zumkeller, Aug 31 2013

Examples

			121 is alternating and in the sequence because its consecutive digits are odd-even-odd, 1 being odd and 2 even. Of course, 1234567890 is also alternating.
		

Links

• Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
• 45th International Mathematical Olympiad (45th IMO), Problem #6 and Solution, Mathematics Magazine, 78 (2005), pp. 247, 250, 251.
• Index entries for 10-automatic sequences.

Crossrefs

Complement: A228709.

Subsequences: A030142, A030143, A030144, A030147, A030152, A062285.

Cf. A110303, A110304, A110305, A056830, A103181, A228722, A228723.

Programs

Haskell

a030141 n = a030141_list !! (n-1)
a030141_list = filter ((== 1) . a228710) [0..]
-- Reinhard Zumkeller, Aug 31 2013

Mathematica

fQ[n_] := Block[{m = Mod[ IntegerDigits@ n, 2]}, m == Split[m, UnsameQ][[1]]]; Select[ Range[0, 130], fQ] (* Robert G. Wilson v, Apr 01 2011 *)
Select[Range[0,150],FreeQ[Differences[Boole[EvenQ[IntegerDigits[#]]]],0]&] (* Harvey P. Dale, Jul 19 2025 *)

PARI

is(n,d=digits(n))=for(i=2,#d, if((d[i]-d[i-1])%2==0, return(0))); 1 \\ Charles R Greathouse IV, Jul 08 2022

Python

from itertools import count
def A030141_gen(startvalue=0): # generator of terms >= startvalue
    return filter(lambda n:all(int(a)+int(b)&1 for a, b in zip(str(n),str(n)[1:])),count(max(startvalue,0)))
A030141_list = list(islice(A030141_gen(),30)) # Chai Wah Wu, Jul 12 2022

Python

from itertools import chain, count, islice
def altgen(seed, digits):
    allowed = "02468" if seed in "13579" else "13579"
    if digits == 1: yield from allowed; return
    for f in allowed: yield from (f + r for r in altgen(f, digits-1))
def agen(): yield from chain(range(10), (int(f+r) for d in count(2) for f in "123456789" for r in altgen(f, d-1)))
print(list(islice(agen(), 65))) # Michael S. Branicky, Jul 12 2022

Extensions

Offset corrected by Reinhard Zumkeller, Aug 31 2013