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a(n) = position of n in the lexicographical ordering A119589 of natural numbers from 1 to 100.
+20
3
1, 13, 24, 35, 46, 57, 68, 79, 90, 2, 4, 5, 6, 7, 8, 9, 10, 11, 12, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 69, 70, 71, 72, 73, 74, 75, 76, 77, 78, 80, 81, 82, 83, 84, 85, 86, 87, 88, 89, 91, 92, 93, 94, 95, 96, 97, 98, 99, 100, 3
FORMULA
a(n) = a(n-1) + a(n-10) - a(n-11) for 21 < n < 100. - M. F. Hasler, Sep 03 2018
EXAMPLE
a(1) = 1
a(10) = 2 because "10" comes after "1"
a(100) = 3 because "100" comes after "10", but before "11"
PROG
(PARI) vecsort(vecsort(vector(100, n, Str(n)), , 1), , 1) \\ M. F. Hasler, Sep 03 2018, simplified Oct 25 2019
CROSSREFS
Cf. A119589 (integers 1..100 in lexicographical order). Cf. A190016, A190017 (integers 1..10^4 in lexicographical order, and inverse).
Numbers 1 through 10000 sorted lexicographically in decimal representation.
+10
10
1, 10, 100, 1000, 10000, 1001, 1002, 1003, 1004, 1005, 1006, 1007, 1008, 1009, 101, 1010, 1011, 1012, 1013, 1014, 1015, 1016, 1017, 1018, 1019, 102, 1020, 1021, 1022, 1023, 1024, 1025, 1026, 1027, 1028, 1029, 103, 1030, 1031, 1032, 1033, 1034, 1035, 1036
COMMENTS
there are 11 fixed points: {1,9980,9981,9982,9983,9984,9985,9986,9987,9988,9989}.
EXAMPLE
a(13) = 1008;
a(14) = 1009;
a(15) = 101;
a(16) = 1010;
a(17) = 1011;
largest term a(5) = 10000;
last term a(10000) = 9999, largest term lexicographically.
PROG
(Haskell)
import Data.Ord (comparing)
import Data.List (sortBy)
a190016 n = a190016_list !! (n-1)
a190016_list = sortBy (comparing show) [1..10000]
(PARI) eval(Set(vector(10^4, n, Str(n)))) \\ M. F. Hasler, Oct 25 2019
Inverse permutation to A190016: lexicographical ordering of integers 1 .. 10^4.
+10
5
1, 1113, 2224, 3335, 4446, 5557, 6668, 7779, 8890, 2, 114, 225, 336, 447, 558, 669, 780, 891, 1002, 1114, 1225, 1336, 1447, 1558, 1669, 1780, 1891, 2002, 2113, 2225, 2336, 2447, 2558, 2669, 2780, 2891, 3002, 3113, 3224, 3336, 3447, 3558, 3669, 3780, 3891
PROG
(Haskell)
import Data.List (elemIndex)
import Data.Maybe (fromJust)
a190017 n = a190017_list !! (n-1)
a190017_list =
map (succ . fromJust . (`elemIndex` a190016_list)) [1..10000]
CROSSREFS
Cf. A190016 (inverse: integers 1..10^4 in lexicographical order). Cf. A119589, A119590 (integers 1..100 in lexicographical order, and inverse).
Number subsets {0, ..., 10^k - 1} written in base 10 and sorted lexicographically, for k = 1, 2, ...
+10
2
0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 0, 1, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 2, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 3, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 4, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 5, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 6, 60, 61
COMMENTS
The sequence is the flattened form of an irregular table T(k, i). The rows for k >= 1 contain a permutation of the numbers 0 <= i <= 10^k - 1 which is defined by the lexicographical order of the numbers i written in base 10.
This "useless" order appears, for example, in a directory listing of numbered filenames, or after an ASCII sort of signatures of linear recurrences. The Perl program in the link computes this sequence and variations with different ranges and bases.
LINKS
Georg Fischer, Perl program which generates this sequence and its inverse.
EXAMPLE
Table T(k, i) begins:
k\i 0 1 2 3 ...
-------------------------
1: 0 1 2 3 ... 9
2: 0 1 10 11 ... 19 2 20 21 ... 99
3: 0 1 10 100 ... 109 11 110 111 ... 999
4: ...
CROSSREFS
Cf. A119589 (like row k=2, but 1 <= i <= 100), A190016 (like row k=4, but 1 <= i <= 10000), A309590 (inverse)
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