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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.

%I A131744 #32 Jan 14 2023 03:08:46
%S A131744 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,
%T A131744 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,
%U A131744 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
%N 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.
%C A131744 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
%C A131744 From _David Applegate_, Sep 24 2007: (Start)
%C A131744 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:
%C A131744 0 -> 0,1,4,5,7,9,10,12,15,21
%C A131744 1 -> 1,3,5,9,12
%C A131744 2 -> 1,3,12,15
%C A131744 3 -> 0,1,2,3,4,5,8,9,11,12,13,14,18
%C A131744 4 -> 2,3,12,14
%C A131744 5 -> 3,5,9,10,12,14,15
%C A131744 6 -> 3,5,12,15,21
%C A131744 7 -> 7,10,17
%C A131744 8 -> 0,3,5,9
%C A131744 9 -> 0,1,2,3,4,5,6,8,9,10,12,14,15,21
%C A131744 10 -> 1,13,15,17
%C A131744 11 -> 21
%C A131744 12 -> 0,1,6,9,10,14,15,21
%C A131744 13 -> 0,3,17
%C A131744 14 -> 3,10,15,17
%C A131744 15 -> 0,3,4,9,12,15,18
%C A131744 17 -> 1,9,10,14,15,17,21
%C A131744 18 -> 3,7,9
%C A131744 21 -> 13,21
%C A131744 (End)
%C A131744 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
%C A131744 The name of this sequence ("Eric Angelini's ... puzzle") was added by _N. J. A. Sloane_ many months after Eric Angelini submitted it.
%C A131744 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
%H A131744 N. J. A. Sloane, <a href="/A131744/b131744.txt">Table of n, a(n) for n = 1..40000</a>
%H A131744 N. J. A. Sloane, <a href="https://arxiv.org/abs/2301.03149">"A Handbook of Integer Sequences" Fifty Years Later</a>, arXiv:2301.03149 [math.NT], 2023, p. 9.
%e A131744 O.N.E...N.I.N.E...N.I.N.E...F.I..V..E...F.I..V..E...
%e A131744 .1.9..9..5.5.9..9..5.5.9..1..3.13.17..1..3.13.17....
%e A131744 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
%t A131744 Nest[Abs@Differences@Flatten[LetterNumber[Characters[IntegerName@#]/."-"->""]&/@#]&,{1},4] (* _Giorgos Kalogeropoulos_, Apr 11 2021 *)
%o A131744 (Python)
%o A131744 def chrdist(a, b): return abs(ord(a)-ord(b))
%o A131744 def aupto(nn):
%o A131744   allnames = "zero,one,two,three,four,five,six,seven,eight,nine,ten,eleven,twelve,thirteen,fourteen,fifteen,sixteen,seventeen,eighteen,nineteen,twenty,twentyone"
%o A131744   names = allnames.split(",")
%o A131744   alst, aidx, last, nxt = [1, 9], 1, "e", "one"
%o A131744   while len(alst) < nn:
%o A131744     nxt = names[alst[aidx]]
%o A131744     alst += [chrdist(a, b) for a, b in zip(last+nxt[:-1], nxt)]
%o A131744     last, aidx = nxt[-1], aidx + 1
%o A131744   return alst[:nn]
%o A131744 print(aupto(84)) # _Michael S. Branicky_, Jan 09 2021
%Y A131744 Cf. A131745, A131746, A130316, A133152, A133816, A133817.
%Y A131744 Cf. A131285 (ranks of letters), A131286, A131287.
%K A131744 nonn,word,nice
%O A131744 1,2
%A A131744 _Eric Angelini_, Sep 20 2007
%E A131744 More terms from _N. J. A. Sloane_, Sep 20 2007