A118757 -id:A118757 - cp's OEIS frontend

A118763 a(n) = smallest number not occurring earlier having in decimal representation to its predecessor Levenshtein distance = 1; a(0)=0.

0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 19, 10, 11, 12, 13, 14, 15, 16, 17, 18, 28, 20, 21, 22, 23, 24, 25, 26, 27, 29, 39, 30, 31, 32, 33, 34, 35, 36, 37, 38, 48, 40, 41, 42, 43, 44, 45, 46, 47, 49, 59, 50, 51, 52, 53, 54, 55, 56, 57, 58, 68, 60, 61, 62, 63, 64, 65, 66, 67, 69, 79, 70, 71

Offset: 0

Reinhard Zumkeller, May 01 2006

Permutation of the natural numbers; inverse: A118764; A118765(n)=a(a(n)); a(A118767(n))=A118767(n);

A118768(n) = a(n+1) - a(n);

Robert G. Wilson v, Table of n, a(n) for n = 0..10000
Michael Gilleland, Levenshtein Distance
Index entries for sequences that are permutations of the natural numbers

Cf. A118757.

levenshtein[s_List, t_List] := Module[{d, n = Length@ s, m = Length@ t}, Which[s === t, 0, n == 0, m, m == 0, n, s != t, d = Table[0, {m + 1}, {n + 1}]; d[[1, Range[n + 1]]] = Range[0, n]; d[[Range[m + 1], 1]] = Range[0, m]; Do[ d[[j + 1, i + 1]] = Min[d[[j, i + 1]] + 1, d[[j + 1, i]] + 1, d[[j, i]] + If[ s[[i]] === t[[j]], 0, 1]], {j, m}, {i, n}]; d[[ -1, -1]] ]]; f[lst_] :=  Block[{k = 1, l = IntegerDigits[ lst[[-1]]]}, While[ MemberQ[lst, k] || levenshtein[l, IntegerDigits[k]] > 1, k++]; Append[lst, k]]; Nest[f, {0}, 100] (* Robert G. Wilson v, Sep 22 2016 *)

from itertools import islice
from Levenshtein import distance as Ld
def agen(): # generator of terms
    an, aset, mink = 0, {0}, 1
    while True:
        yield an
        s, k = str(an), mink
        while k in aset or Ld(s, str(k)) != 1: k += 1
        an = k
        aset.add(k)
        while mink in aset: mink += 1
print(list(islice(agen(), 73))) # Michael S. Branicky, Dec 01 2023

A003100 Decimal Gray code for n.

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 59, 58, 57, 56, 55, 54, 53, 52, 51, 50, 60, 61, 62, 63, 64, 65, 66, 67, 68, 69, 79, 78, 77

Offset: 0

N. J. A. Sloane, Henry W. Gould

This permutation of the nonnegative integers is not self-inverse, as previously claimed. The first exception is a(100) = 190, but a(190) = 109. - Franklin T. Adams-Watters, Mar 05 2010

a(n) = A118757(n) for n<=100, = a(100)=A118757(100)=190, but a(101)=191, A118757(101)=180. - Reinhard Zumkeller, May 01 2006

M. Gardner, Knotted Doughnuts and Other Mathematical Entertainments. Freeman, NY, 1986, p. 18.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Franklin T. Adams-Watters, Table of n, a(n) for n = 0..10000
A. J. Cole, Cyclic progressive number systems, Math. Gaz., 50 (1966), 122-131.
A. J. Cole, Cyclic progressive number systems, Math. Gaz., 50 (1966), 122-131. [Annotated scanned copy]
Henry W. Gould, Letters to N. J. A. Sloane, Oct 1973 and Jan 1974.
D. E. Knuth, Gray Codes (Vol. 4 of TAOCP)
Index entries for sequences that are permutations of the natural numbers

Inverse is A174025.

A003100 :=proc(n)
    local s,i:
    s:=[op(convert(n,base,10)),0]:
    add(piecewise(s[i+1] mod 2=0,s[i],9-s[i])*10^(i-1),i=1..nops(s)-1) :
end proc:
seq(A003100(j),j=0..100); # Pab Ter, Oct 14 2005

More terms from Pab Ter (pabrlos2(AT)yahoo.com), Oct 14 2005

Incorrect comment replaced by Franklin T. Adams-Watters, Mar 05 2010

A118760 A118758(A118758(n)).

Original entry on oeis.org

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, 118, 108, 127, 126

Offset: 0

Reinhard Zumkeller, May 01 2006

nonn

Inverse integer permutation of A118759; a(n)=A118759(n) for n<100; A118757(a(n))=a(A118757(n))=A118758(n).

Tanya Khovanova, Non Recursions
Index entries for sequences that are permutations of the natural numbers

Cf. A118759.

A261725 Lexicographically earliest sequence of distinct terms such that the absolute difference of two successive terms is a power of 10, and can be computed without carry.

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 59, 58, 57, 56, 55, 54, 53, 52, 51, 50, 60, 61, 62, 63, 64, 65, 66, 67

Offset: 0

Paul Tek, Aug 30 2015

In base 10, two successive terms have the same representation, except for one position, where the digits differ from exactly one unit. This difference can occur on a leading zero.
Conjectured to be a permutation of the nonnegative integers. See A261729 for putative inverse.
a(n) = A003100(n) for n < 101, but a(101) = 180, A003100(101) = 191.
a(n) = A118757(n) for n < 201, but a(201) = 281, A118757(201) = 290.
a(n) = A118758(n) for n < 100, but a(100) = 190, A118758(100) = 109.
a(n) = A174025(n) for n < 100, but a(100) = 190, A174025(100) = 199.
a(n) = A261729(n) for n < 100, but a(100) = 190, A261729(100) = 109.

Paul Tek, Table of n, a(n) for n = 0..10000
Paul Tek, PERL program for this sequence

Cf. A003100, A118757, A118763, A163252, A261729 (putative inverse).

Perl
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See Links section.
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A261729 Putative inverse of conjectured permutation in A261725.

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 59, 58, 57, 56, 55, 54, 53, 52, 51, 50, 60, 61, 62, 63, 64, 65, 66, 67

Offset: 0

Paul Tek, Aug 30 2015

a(n) = A003100(n) for n < 100, but a(100) = 109, A003100(100) = 190.
a(n) = A118757(n) for n < 100, but a(100) = 109, A118757(100) = 190.
a(n) = A118758(n) for n < 201, but a(201) = 209, A118758(201) = 211.
a(n) = A174025(n) for n < 100, but a(100) = 109, A174025(100) = 199.
a(n) = A261725(n) for n < 100, but a(100) = 109, A261725(100) = 190.

Paul Tek, Table of n, a(n) for n = 0..10000

Cf. A261725.

A367638 Sequence S of positive integers such that the successive digits d of S are the successive Levenshtein distances between two adjacent terms of S. When possible, S is always extended with the smallest positive integer not yet present.

Original entry on oeis.org

1, 2, 10, 11, 11, 12, 13, 3, 4, 5, 14, 15, 200, 6, 1000, 22111, 2111, 7, 8, 10000, 100, 100, 100, 222211, 22211, 22211, 22211, 22211, 211, 16, 17, 18, 19, 20, 21, 22, 23, 1000000, 22111111, 2111111, 2111111, 2111111, 2111111, 2111111, 111111, 111111, 111111, 11111, 11111, 11111, 1111, 1111, 1111, 101, 30, 9, 24

Offset: 1

Eric Angelini and Giorgos Kalogeropoulos, Nov 25 2023

			The sequence starts with 1, 2, 10, 11, 11, 12, 13, 3.
a(1) = 1 is indeed the Ld (Levenshtein distance) between a(1) = 1 and a(2) = 2;
a(2) = 2 is the Ld between a(2) = 2 and a(3) = 10;
a(3) = 10 whose first digit 1 is the Ld between a(3) = 10 and a(4) = 11;
a(3) = 10 whose second digit 0 is the Ld between a(4) = 11 and a(5) = 11;
a(4) = 11 whose first digit 1 is the Ld between a(5) = 11 and a(6) = 12;
a(4) = 11 whose second digit 1 is the Ld between a(6) = 12 and a(7) = 13;
a(5) = 11 whose first digit 1 is the Ld between a(7) = 13 and a(8) = 3; etc.

Éric Angelini, A sequence with Vladimir Iosifovich (and his wife), Personal blog, Nov 2023.

Cf. A118757, A151875.

a[1]=1;a[n_]:=a[n]=If[Flatten[IntegerDigits/@(ar=Array[a,n-1])][[n-1]]==0,a[n-1],(k=1;While[MemberQ[ar,k]||EditDistance[ToString@a[n-1],ToString@k]!=Flatten[IntegerDigits/@Join[ar,{k}]][[n-1]],k++];k)];Array[a,23]

cp's OEIS Frontend

A118758 Inverse of A118757.

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A118761 Fixed points of permutations A118757, A118758, A118759 and A118760.

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A118759 A118757(A118757(n)).

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A118762 First differences of A118757.

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A118763 a(n) = smallest number not occurring earlier having in decimal representation to its predecessor Levenshtein distance = 1; a(0)=0.

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A003100 Decimal Gray code for n.

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A118760 A118758(A118758(n)).

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A261725 Lexicographically earliest sequence of distinct terms such that the absolute difference of two successive terms is a power of 10, and can be computed without carry.

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Perl

A261729 Putative inverse of conjectured permutation in A261725.

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A367638 Sequence S of positive integers such that the successive digits d of S are the successive Levenshtein distances between two adjacent terms of S. When possible, S is always extended with the smallest positive integer not yet present.

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