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A282636 a(1) = 1. a(n) is the smallest natural number such that a(n) > a(n-1), the name of a(n) in (American) English ends with the letter a(n-1) starts with, and a(n) makes the sequence extendable.

%I A282636 #7 Feb 23 2017 22:59:41
%S A282636 1,2,8,9,10,28,38,88,89,91,97,107,122,132,142,152,162,172,182,192,202,
%T A282636 208,228,238,248,258,268,278,288,298,308,328,338,348,358,368,378,388,
%U A282636 398,808,809,812,821,823,825,829,831,833,835,839,841,843,845,849,851,853,855,859
%N A282636 a(1) = 1. a(n) is the smallest natural number such that a(n) > a(n-1), the name of a(n) in (American) English ends with the letter a(n-1) starts with, and a(n) makes the sequence extendable.
%C A282636 "a(n) makes the sequence extendable" means the name of a(n) does not start with any of the letters f and s.
%e A282636 38 is followed by 88 as 48, 58, 68 and 78 cannot make the sequence extendable.
%t A282636 names = StringReplace[
%t A282636    IntegerName /@ Range[1000], {"\[Hyphen]" -> "", " " -> ""}];
%t A282636 lst = {1}; next[n_] := next[n] =
%t A282636   Module[{pos = Last[lst] + 1},
%t A282636    While[
%t A282636     Or[
%t A282636      And[First[Characters[names[[Last[lst]]]]] !=
%t A282636        Last[Characters[names[[pos]]]], pos <= Length[names]],
%t A282636      MemberQ[{"f", "s"}, First[Characters[names[[pos]]]]] == True
%t A282636      ],
%t A282636     pos++
%t A282636     ];
%t A282636    AppendTo[lst, pos]
%t A282636    ];
%t A282636 seq[n_] := Nest[next, 1, n]; seq[57]
%Y A282636 Cf. A277971, A277972, A281067, A281068, A282637.
%K A282636 easy,nonn,word
%O A282636 1,2
%A A282636 _Ivan N. Ianakiev_, Feb 20 2017