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A009994
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Numbers with digits in nondecreasing order.
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66
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0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 11, 12, 13, 14, 15, 16, 17, 18, 19, 22, 23, 24, 25, 26, 27, 28, 29, 33, 34, 35, 36, 37, 38, 39, 44, 45, 46, 47, 48, 49, 55, 56, 57, 58, 59, 66, 67, 68, 69, 77, 78, 79, 88, 89, 99, 111, 112, 113, 114, 115, 116, 117, 118, 119, 122
(list;
graph;
refs;
listen;
history;
text;
internal format)
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OFFSET
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1,3
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COMMENTS
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This sequence was used by the U.S. Bureau of the Census in the mid-1950s to numerically code the alphabetical list of counties within a state (with some modifications for Texas). The 3-digit code has a "self-policing element" built into it and "was fairly effective in detecting the transposition of 2 digits." (Hanna 1959). - Randy A. Becker, Dec 11 2017
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REFERENCES
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Amarnath Murthy and Robert J. Clarke, Some Properties of Staircase sequence, Mathematics & Informatics Quarterly, Volume 11, No. 4, November 2001.
Frank A. Hanna, The Compilation of Manufacturing Statistics. U.S. Department of Commerce, Bureau of the Census, 1959.
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LINKS
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David Applegate, Marc LeBrun, N. J. A. Sloane, Dismal Arithmetic, J. Int. Seq. 14 (2011) # 11.9.8.
David Radcliffe and Brendan McKay, Re: A009994, SeqFan list, Jul 29 2019
Eric Weisstein's World of Mathematics, Digit.
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FORMULA
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a(n) ~ 10^((9! n)^(1/9) - 5), since 10^(d - 1) <= a(n) < 10^d for binomial(d + 8, 9) < n <= binomial(d + 9, 9) = (d + 5 - epsilon)^9 / 9!. Using epsilon = 10/(3n) + o(1/n) gives more precise estimate. [Following Radcliffe and McKay, cf. SeqFan list.] - M. F. Hasler, Jul 30 2019
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MAPLE
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A[0]:= [0]: A[1]:= [$1..9]:
for d from 2 to 4 do
A[d]:= map(t -> seq(10*t+i, i=(t mod 10) .. 9), A[d-1]):
od:
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MATHEMATICA
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Select[Range[0, 125], LessEqual@@IntegerDigits[#] &] (* Ray Chandler, Oct 25 2011 *)
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PROG
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(Haskell)
import Data.Set (fromList, deleteFindMin, insert)
a009994 n = a009994_list !! n
a009994_list = 0 : f (fromList [1..9]) where
f s = m : f (foldl (flip insert) s' $ map (10*m +) [m `mod` 10 ..9])
where (m, s') = deleteFindMin s
(Python)
from itertools import combinations_with_replacement
def A009994generator():
yield 0
l = 1
while True:
for i in combinations_with_replacement('123456789', l):
yield int(''.join(i))
(Magma) [k:k in [0..122]|Sort(Intseq(k)) eq Reverse(Intseq(k))]; // Marius A. Burtea, Jul 28 2019
(Scala) def hasDigitsSorted(n: Int): Boolean = {
val digSort = Integer.parseInt(n.toString.toCharArray.sorted.mkString)
n == digSort
}
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CROSSREFS
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Apart from the first term, a subsequence of A052382. A254143 is a subsequence.
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KEYWORD
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AUTHOR
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STATUS
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approved
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