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A033619 Undulating numbers (of form abababab... in base 10).

%I A033619 #32 Jul 08 2025 19:54:05
%S A033619 0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,
%T A033619 26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,
%U A033619 49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,101,111,121,131,141,151
%N A033619 Undulating numbers (of form abababab... in base 10).
%D A033619 C. A. Pickover, "Keys to Infinity", Wiley 1995, p. 159,160.
%D A033619 C. A. Pickover, "Wonders of Numbers", Oxford New York 2001, Chapter 52, pp. 123-124, 316-317.
%H A033619 Reinhard Zumkeller, <a href="/A033619/b033619.txt">Table of n, a(n) for n = 1..10000</a>
%H A033619 C. A. Pickover, "Wonders of Numbers, Adventures in Mathematics, Mind and Meaning," <a href="http://www.zentralblatt-math.org/zmath/en/search/?q=an:0983.00008&amp;format=complete">Zentralblatt review</a>
%H A033619 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/UndulatingNumber.html">Undulating Number</a>
%p A033619 $0..9,seq(seq(seq(a*(10^(d+1)-10^(d+1 mod 2))/99 + b*(10^d - 10^(d mod 2))/99, b=0..9),a=1..9),d=2..6); # _Robert Israel_, Jul 08 2016
%t A033619 wave[1] = Range[0, 9]; wave[2] = Range[10, 99]; wave[n_] := wave[n] = Select[ Union[ Flatten[ {id = IntegerDigits[#]; FromDigits[ Prepend[id, id[[2]]]], FromDigits[ Append[id, id[[-2]]]]} & /@ wave[n-1]]], 10^(n-1) < # < 10^n & ]; Flatten[ Table[ wave[n], {n, 1, 3}]] (* _Jean-François Alcover_, Jun 19 2012 *)
%o A033619 (Haskell)
%o A033619 import Data.Set (fromList, deleteFindMin, insert)
%o A033619 a033619 n = a033619_list !! (n-1)
%o A033619 a033619_list = [0..9] ++ (f $ fromList [10..99]) where
%o A033619    f s = m : f (insert (m * 10 + h) s') where
%o A033619      h = div (mod m 100) 10
%o A033619      (m,s') = deleteFindMin s
%o A033619 -- _Reinhard Zumkeller_, May 01 2012
%o A033619 (Python)
%o A033619 from itertools import count, islice
%o A033619 def agen(): # generator of terms
%o A033619     yield from range(10)
%o A033619     for d in count(2):
%o A033619         q, r = divmod(d, 2)
%o A033619         for a in "123456789":
%o A033619             for b in "0123456789":
%o A033619                 yield int((a+b)*q + a*r)
%o A033619 print(list(islice(agen(), 106))) # _Michael S. Branicky_, Mar 28 2022
%Y A033619 Cf. A016073, A046075, A122875, A242541.
%K A033619 nonn,base,easy
%O A033619 1,3
%A A033619 _N. J. A. Sloane_