cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

User: A. D. Skovgaard

A. D. Skovgaard's wiki page.

A. D. Skovgaard has authored 4 sequences.

A308831 Start with generation 0, which is the empty sequence. For generation N>=1, extend the existing sequence into a non-cyclic ternary de Bruijn sequence of order N. If more than one extension is possible, choose the lexicographically earliest.

Original entry on oeis.org

0, 1, 2, 0, 0, 2, 2, 1, 1, 0, 0, 0, 1, 0, 1, 1, 1, 2, 1, 0, 2, 0, 2, 1, 2, 2, 2, 0, 1, 0, 0, 1, 1, 0, 1, 0, 2, 1, 0, 0, 2, 0, 0, 0, 0, 2, 1, 1, 1, 1, 0, 2, 2, 0, 0, 1, 2, 1, 1, 2, 0, 1, 1, 2, 2, 0, 2, 0, 1, 2, 2, 1, 2, 1, 2, 0, 2, 2, 2, 2, 1, 0, 1, 2
Offset: 0

Views

Author

A. D. Skovgaard, Jun 27 2019

Keywords

Comments

If using a binary alphabet instead, it would not be possible to extend the sequence infinitely as a de Bruijn sequence (order 3 needs an extra term: 01100010111). - A. D. Skovgaard, Apr 19 2020

Examples

			Generation 1:
[012] (All ternary sequences of length 1 now appear. With 3! = 6 solutions, the lexicographically earliest is chosen.)
Generation 2:
[0120022110] (The sequence is extended from the previous generation, now including all ternary sequences of length 2.)
The process continues.

Links

Crossrefs

Cf. A080679 (binary equivalent), A166315, A169676.

A306211 Concatenation of the current sequence with the lengths of the runs in the sequence, with a(1) = 1.

Original entry on oeis.org

1, 1, 2, 2, 1, 2, 2, 1, 2, 2, 1, 2, 1, 2, 2, 1, 2, 1, 2, 1, 1, 1, 2, 2, 1, 2, 1, 2, 1, 1, 1, 2, 1, 1, 1, 1, 3, 2, 2, 1, 2, 1, 2, 1, 1, 1, 2, 1, 1, 1, 1, 3, 2, 1, 1, 1, 1, 3, 1, 4, 1, 2, 2, 1, 2, 1, 2, 1, 1, 1, 2, 1, 1, 1, 1, 3, 2, 1, 1, 1, 1, 3, 1, 4, 1, 2, 1, 1, 1, 1, 3
Offset: 1

Views

Author

A. D. Skovgaard, Jan 29 2019

Keywords

Comments

Conjecture: All terms are less than or equal to 5. - Peter Kagey, Jan 29 2019
Conjecture: Every number appears! (Based on the analogy with the somewhat similar sequence A090822, where the first 5 appeared at around 10^(10^23) steps). - N. J. A. Sloane, Jan 29 2019
An alternative definition: Start with 1, extend the sequence by appending its RUNS transform, recompute the RUNS transform, append it, repeat. - N. J. A. Sloane, Jan 29 2019
The first time we see 1, 2, 3, 4, 5 is at n=1, 3, 37, 60, 255 (A323829). After 65 generations (10228800161220 terms) the largest term is 5. The relative frequencies of 1..5 are roughly 0.71, 6.7e-9, 0.23, 1.6e-8, 0.061. 2s and 4s appear to get rarer as n increases. - Benjamin Chaffin, Feb 07 2019
If we record the successive RUNS transforms and concatenate them, we get 1; 2; 2, 1; 2, 2, 1; 2, 2, 1, 2, 1; ..., which is this sequence without the initial 1. - A. D. Skovgaard, Jan 30 2019 (Rephrased by N. J. A. Sloane, Jan 30 2019)

Examples

			a(2) = 1, since there is a run of length 1 at a(1).
a(3) = 2, since there is a run of length 2 at a(1..2).
a(4..5) = 2, 1, since the runs are as follows:
  1, 1, 2  a(1..3)
  \__/  |
  2,    1  a(4..5)
a(37) = 3, since a(20..22) = 1, 1, 1.
Steps in construction:
  [1]  initial sequence
  [1]  its run length
 .
  [1, 1] concatenation of above is new sequence
  [2]  its run length
 .
  [1, 1, 2] concatenation of above is new sequence
  [2, 1]  its run lengths
 .
  [1, 1, 2, 2, 1]
  [2, 2, 1]
 .
  [1, 1, 2, 2, 1, 2, 2, 1]
  [2, 2, 1, 2, 1]
 .
  [1, 1, 2, 2, 1, 2, 2, 1, 2, 2, 1, 2, 1]
  [2, 2, 1, 2, 1, 2, 1, 1, 1]
 .
  [1, 1, 2, 2, 1, 2, 2, 1, 2, 2, 1, 2, 1, 2, 2, 1, 2, 1, 2, 1, 1, 1]
  [2, 2, 1, 2, 1, 2, 1, 1, 1, 2, 1, 1, 1, 1, 3]
 .
  [1, 1, 2, 2, 1, 2, 2, 1, 2, 2, 1, 2, 1, 2, 2, 1, 2, 1, 2, 1, 1, 1, 2, 2, 1, 2, 1, 2, 1, 1, 1, 2, 1, 1, 1, 1, 3]
From _N. J. A. Sloane_, Jan 31 2019: (Start)
The first 9 generations, in compressed notation (see A323477) are:
  1
  11
  112
  11221
  11221221
  1122122122121
  1122122122121221212111
  1122122122121221212111221212111211113
  1122122122121221212111221212111211113221212111211113211113141
  ... (End)

Links

Crossrefs

Cf. A000002, A107946, A306215, A090822.
Positions of 3's, 4's, 5's: A323476, A306222, A306223.
Successive generations: A323477, A323478, A306215, A323475, A306333.
See also A323479, A323480, A323481, A323826 (RUNS transform), A323827, A323829 (where n first appears).

Programs

  • Haskell
    group [] = []
group (x:xs)= (x:ys):group zs where (ys,zs) = span (==x) xs
a306211_next_gen xs = xs ++ (map length $ group xs)
a306211_gen 1 = [1]
a306211_gen n = a306211_next_gen $ a306211_gen (n-1)
a306211 n = a306211_gen n !! (n-1)
-- Jean-François Antoniotti, Jan 31 2021
  • Mathematica
    seq[n_] := seq[n] = If[n==1, {1}, Join[seq[n-1], Length /@ Split[seq[n-1]]]];
seq[10] (* Jean-François Alcover, Jul 19 2022 *)

A274206 a(n) = the last nonzero digit of n followed by all the trailing zeros of n.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 1, 2, 3, 4, 5, 6, 7, 8, 9, 20, 1, 2, 3, 4, 5, 6, 7, 8, 9, 30, 1, 2, 3, 4, 5, 6, 7, 8, 9, 40, 1, 2, 3, 4, 5, 6, 7, 8, 9, 50, 1, 2, 3, 4, 5, 6, 7, 8, 9, 60, 1, 2, 3, 4, 5, 6, 7, 8, 9, 70, 1, 2, 3, 4, 5, 6, 7, 8, 9, 80
Offset: 1

Views

Author

A. D. Skovgaard, Jun 13 2016

Keywords

Comments

a(n) is the number formed by the rightmost A160094(n) digits -- only the position(s) that changed -- of a decimal counter (e.g., an odometer) after it increments from n - 1 to n. - Rick L. Shepherd, Jun 29 2016

Examples

			a(1) = 1 because when 1 is added to 1 - 1 = 0, the units digit changes so the units digit of 1 is shown.
a(110) = 10 because when 1 is added to 109, the tens digit and the units digit change, so the last two digits of 110 are shown.
a(1000) = 1000 because when 1 is added to 999, all the digits change so they are all shown.

Links

Crossrefs

Cf. A010879, A037124 (these increasing distinct terms), A006519 (binary equivalent shown in decimal), A160094.

Programs

  • Maple
    f:= n -> n mod 10^(1+min(padic:-ordp(n,2), padic:-ordp(n,5))):
map(f, [$1..100]); # Robert Israel, Aug 08 2016
  • Mathematica
    Table[FromDigits@ Join[{Last@ #}, Table[0, {Log10[n/FromDigits@ #]}]] &@ Select[IntegerDigits@ n, # != 0 &], {n, 120}] (* Michael De Vlieger, Jun 29 2016 *)
  • PARI
    a(n) = n%10^(valuation(n,10)+1); \\ David A. Corneth, Jun 29 2016

Formula

a(n) = n mod 10 if n is not a multiple of 10.
From Robert Israel, Aug 08 2016: (Start)
a(10*n) = 10*a(n).
a(10*n+k) = k for 1 <= k <= 9.
G.f. g(x) satisfies g(x) = (x+2x^2+...+9x^9)/(1-x^10) + 10 g(x^10). (End)

A272264 Numbers that become a different number when flipped upside down.

Original entry on oeis.org

6, 9, 16, 18, 19, 61, 66, 68, 81, 86, 89, 91, 98, 99, 106, 108, 109, 116, 118, 119, 161, 166, 168, 169, 186, 188, 189, 191, 196, 198, 199, 601, 606, 608, 611, 616, 618, 661, 666, 668, 669, 681, 686, 688, 691, 696, 698, 699, 801, 806, 809, 811, 816, 819, 861, 866, 868, 869, 881, 886, 889
Offset: 1

Views

Author

A. D. Skovgaard, Apr 24 2016

Keywords

Comments

Although 2 and 5 flipped upside down on a digital clock are numbers, they are not permitted here. - David A. Corneth, May 22 2016

Links

Crossrefs

Cf. A000787, A045574.

Programs

  • PARI
    is(n) = {my(d=digits(n),dr); if(d[#d]==0 || #setminus(Set(d),Set([0,1,6,8,9])) !=0, return(0), dr=vector(#d)); for(i=1,#d, dr[#d-i+1] = if(d[i]==6||d[i]==9,15-d[i],d[i]));dr!=d} \\ David A. Corneth, May 22 2016

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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