A131744 Eric Angelini's "1995" puzzle: the sequence is defined by the property that if one writes the English names for the entries, replaces each letter with its rank in the alphabet and calculates the absolute values of the differences, one recovers the sequence.

1, 9, 9, 5, 5, 9, 9, 5, 5, 9, 1, 3, 13, 17, 1, 3, 13, 17, 9, 5, 5, 9, 9, 5, 5, 9, 1, 3, 13, 17, 1, 3, 13, 17, 9, 5, 5, 9, 10, 1, 9, 15, 12, 10, 13, 0, 15, 12, 1, 9, 2, 15, 0, 9, 5, 14, 17, 17, 9, 6, 15, 0, 9, 1, 1, 9, 15, 12, 10, 13, 0, 15, 12, 1, 9, 2, 15, 0, 9, 5, 14, 17, 17, 9

Offset: 1

Author

Eric Angelini, Sep 20 2007

Keywords

• nonn
• word
• nice

Comments

In the first few million terms, the numbers 16, 19, 20 and 22-26 do not occur. Of the numbers that do occur, the number 11 appears with the smallest frequence - see A133152. - N. J. A. Sloane, Sep 22 2007

From David Applegate, Sep 24 2007: (Start)

The numbers 16, 19-20, 22-25 never occur in the sequence. The following table gives the possible numbers that can occur in the sequence and for each one, the possible numbers that can follow it. The table is complete - when any number and its successor are expanded, the resulting pairs are also in the table. It contains the expansion of 1 and thus describes all possible transitions:

0 -> 0,1,4,5,7,9,10,12,15,21

1 -> 1,3,5,9,12

2 -> 1,3,12,15

3 -> 0,1,2,3,4,5,8,9,11,12,13,14,18

4 -> 2,3,12,14

5 -> 3,5,9,10,12,14,15

6 -> 3,5,12,15,21

7 -> 7,10,17

8 -> 0,3,5,9

9 -> 0,1,2,3,4,5,6,8,9,10,12,14,15,21

10 -> 1,13,15,17

11 -> 21

12 -> 0,1,6,9,10,14,15,21

13 -> 0,3,17

14 -> 3,10,15,17

15 -> 0,3,4,9,12,15,18

17 -> 1,9,10,14,15,17,21

18 -> 3,7,9

21 -> 13,21

(End)

The sequence may also be extended in the reverse direction: ... 0 21 21 13 3 0 [then what we have now] 1 9 9 5 5 ..., corresponding to ... zero twentyone twentyone thirteen three zero one nine nine five ... - N. J. A. Sloane, Sep 27 2007

The name of this sequence ("Eric Angelini's ... puzzle") was added by N. J. A. Sloane many months after Eric Angelini submitted it.

Begin with 1, map the integer to its name and then map according to A073029, compute the absolute difference, spell out that difference; iterate as necessary. - Robert G. Wilson v, Jun 08 2010

Examples

			O.N.E...N.I.N.E...N.I.N.E...F.I..V..E...F.I..V..E...
.1.9..9..5.5.9..9..5.5.9..1..3.13.17..1..3.13.17....
1 -> "one" -> 15,14,5 -> (the difference is) 1,9; iterate. Therefore 1,9 -> "one,nine"; -> 15,14,5,14,9,14,5 -> 1,9,9,5,5,9; "one,nine,nine,five,five,nine"; etc. - _Robert G. Wilson v_, Jun 08 2010
		

Links

• N. J. A. Sloane, Table of n, a(n) for n = 1..40000
• N. J. A. Sloane, "A Handbook of Integer Sequences" Fifty Years Later, arXiv:2301.03149 [math.NT], 2023, p. 9.

Crossrefs

Cf. A131745, A131746, A130316, A133152, A133816, A133817.

Cf. A131285 (ranks of letters), A131286, A131287.

Programs

Mathematica

Nest[Abs@Differences@Flatten[LetterNumber[Characters[IntegerName@#]/."-"->""]&/@#]&,{1},4] (* Giorgos Kalogeropoulos, Apr 11 2021 *)

Python

def chrdist(a, b): return abs(ord(a)-ord(b))
def aupto(nn):
  allnames = "zero,one,two,three,four,five,six,seven,eight,nine,ten,eleven,twelve,thirteen,fourteen,fifteen,sixteen,seventeen,eighteen,nineteen,twenty,twentyone"
  names = allnames.split(",")
  alst, aidx, last, nxt = [1, 9], 1, "e", "one"
  while len(alst) < nn:
    nxt = names[alst[aidx]]
    alst += [chrdist(a, b) for a, b in zip(last+nxt[:-1], nxt)]
    last, aidx = nxt[-1], aidx + 1
  return alst[:nn]
print(aupto(84)) # Michael S. Branicky, Jan 09 2021

Extensions

More terms from N. J. A. Sloane, Sep 20 2007