A124172 Triangle in which row n contains values from 1 to n in alphabetical order (in English).

1, 1, 2, 1, 3, 2, 4, 1, 3, 2, 5, 4, 1, 3, 2, 5, 4, 1, 6, 3, 2, 5, 4, 1, 7, 6, 3, 2, 8, 5, 4, 1, 7, 6, 3, 2, 8, 5, 4, 9, 1, 7, 6, 3, 2, 8, 5, 4, 9, 1, 7, 6, 10, 3, 2, 8, 11, 5, 4, 9, 1, 7, 6, 10, 3, 2, 8, 11, 5, 4, 9, 1, 7, 6, 10, 3, 12, 2, 8, 11, 5, 4, 9, 1, 7, 6, 10, 13, 3, 12, 2

Offset: 1

Author

Colm Mulcahy, Dec 05 2006

Keywords

• nonn
• tabl
• word

Comments

This is a fractal sequence: if the first instance of each number is deleted, the original sequence is recovered. - Franklin T. Adams-Watters, Nov 24 2013

While it is unlikely to ever make a practical difference, for the sake of definiteness we specify that standard American number names are to be used. - Franklin T. Adams-Watters, Nov 24 2013

For 100th row, "one"=50th value and "one hundred"=51st value, placing the extremes in the center. - Peter Woodward, Jun 27 2015

We distinguish two kinds of "alphabetical order": letter-by-letter and word-by-word. For example, the a- and b-files in A340671, as well as my Mathematica code below (wherein you can make k any integer up to an arbitrary klimit), assume word-by-word (i.e., a space precedes any letter). For A124172 the assumption first impacts a(319602) = 800 (not 18). Ross Eckler first noted the sequence in the February 1981 issue of Word Ways (see p. 20 in the link), attributing it to Philip Cohen. - Hans Havermann, Aug 13 2024

Examples

			Triangle begins:
  1,
  1,2,
  1,3,2,
  4,1,3,2
  5,4,1,3,2
  5,4,1,6,3,2,
  ...
		

Links

• A. Ross Eckler, Alphabetizing the Integers (Word Ways, 1981, Vol. 14, No. 1, pp. 18-20).

Crossrefs

Cf. A002260, A340671.

Programs

Mathematica

names = {"one", "two", "three", "four", "five", "six", "seven", "eight", "nine", "ten", "eleven", "twelve", "thirteen"}; Flatten[Table[Table[Position[names, Sort[Take[names, n]][[i]], 1, 1][[1, 1]], {i, n}], {n, Length[names]}]] (* T. D. Noe, Nov 25 2013 *)
klimit=100; u=IntegerName[Range[klimit],"Words"]; k=24; seq={}; Do[r=Take[u,i]; s=AlphabeticSort[r]; t=Table[FirstPosition[r,s[[j]]][[1]],{j,i}]; seq=Join[seq,t],{i,k}]; seq (* Hans Havermann, Aug 13 2024 *)

Extensions

Edited name and more terms from Franklin T. Adams-Watters, Nov 24 2013