A122875 -id:A122875 - cp's OEIS frontend

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

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, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 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

Offset: 1

N. J. A. Sloane

C. A. Pickover, "Keys to Infinity", Wiley 1995, p. 159,160.
C. A. Pickover, "Wonders of Numbers", Oxford New York 2001, Chapter 52, pp. 123-124, 316-317.

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
C. A. Pickover, "Wonders of Numbers, Adventures in Mathematics, Mind and Meaning," Zentralblatt review
Eric Weisstein's World of Mathematics, Undulating Number

Cf. A016073, A046075, A122875, A242541.

import Data.Set (fromList, deleteFindMin, insert)
a033619 n = a033619_list !! (n-1)
a033619_list = [0..9] ++ (f $ fromList [10..99]) where
   f s = m : f (insert (m * 10 + h) s') where
     h = div (mod m 100) 10
     (m,s') = deleteFindMin s
-- Reinhard Zumkeller, May 01 2012

$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

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 *)

from itertools import count, islice
def agen(): # generator of terms
    yield from range(10)
    for d in count(2):
        q, r = divmod(d, 2)
        for a in "123456789":
            for b in "0123456789":
                yield int((a+b)*q + a*r)
print(list(islice(agen(), 106))) # Michael S. Branicky, Mar 28 2022

A016073 Undulating squares.

Original entry on oeis.org

0, 1, 4, 9, 16, 25, 36, 49, 64, 81, 121, 484, 676, 69696

Offset: 0

Robert G. Wilson v

Numbers with decimal expansion ababab... which are squares.

"Most mathematicians believe we will never find a larger one" - this has now been proved by David Moews.

C. A. Pickover, "Keys to Infinity", Wiley 1995, pp. 159, 160.
C. A. Pickover, "Wonders of Numbers", Oxford New York 2001, Chapter 52, pp. 123-124, 316-317.
Clifford A. Pickover, A Passion for Mathematics, Wiley, 2005; see p. 68.

D. Moews, Home Page [See the paper "No More Undulating Squares", available in LaTeX, DVI and Postscript]
C. A. Pickover, "Wonders of Numbers, Adventures in Mathematics, Mind and Meaning," Zentralblatt review
Eric Weisstein's World of Mathematics, Undulating Number.

Numbers in A033619 that are squares. See A122875 for the square roots.

select(issqr,[$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

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 &]; A016073 = Reap[Do[Do[wk = wave[n][[k]]; If[IntegerQ[Sqrt[wk]], Sow[wk]], {k, 1, Length[wave[n]]}], {n, 1, 5}]][[2, 1]] (* Jean-François Alcover, Dec 28 2012 *)

cp's OEIS Frontend

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

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