%I #5 Aug 16 2020 12:50:08
%S 9,10,101,1,12,11,2,123,113,22,3,13,4,135,225,33,41,37,44,5,15,7,25,8,
%T 337,414,371,14,49,55,6,151,27,79,251,38,81,137,114,47,31,21,48,491,
%U 45,51,16,67,1132,71,479,112,143,816,812,17,814,84,710,313,215,481,24,154,510,512,161,26,78,129,715,4910
%N The rich sandwiches sequence (see Comments lines for definition).
%C Imagine we would have a pair of adjacent integers in the sequence like [1951, 2020]. The sandwich would then be made of the rightmost digit of a(n), the leftmost digit of a(n+1) and, in between, the sum of those two digits. The pair [1951, 2020] would then produce the (rich) sandwich 132. (Why rich? Because a poor sandwich would insert the absolute difference of the digits instead of their sum  that is 112 in this example). Please note that the pair [2020, 1951] would produce the rich and genuine sandwich 011 (we keep the leading zero: these are sandwiches after all, not integers).
%C Now we want the sequence to be the lexicographically earliest sequence of distinct positive terms such that the successive sandwiches emerging from the sequence rebuild it, digit after digit.
%e The first successive sandwiches are: 9101, 011, 121, 121, 231, 132,...
%e The first one (9101) is visible between a(1) = 9 and a(2) = 10; we get the sandwich by inserting the sum 10 between 9 and 1.
%e The second sandwich (011) is visible between a(2) = 10 and a(3) = 101; we get this sandwich by inserting the sum 1 between 0 and 1.
%e The third sandwich (121) is visible between a(3) = 101 and a(4) = 1; we get this sandwich by inserting the sum 2 between 1 and 1; etc.
%e The successive sandwiches rebuild, digit by digit, the starting sequence.
%Y Cf. A335600 (the poor sandwiches sequence).
%K base,nonn
%O 1,1
%A _Eric Angelini_ and _Carole Dubois_, Jul 31 2020
