%I #9 Sep 10 2023 22:38:39
%S 18,27,36,45,1098,1188,1278,1368,1458,1809,1818,1827,1836,1845,2097,
%T 2187,2277,2367,2457,2709,2718,2727,2736,2745,3096,3186,3276,3366,
%U 3456,3609,3618,3627,3636,3645,4095,4185,4275,4365,4455,4509,4518,4527,4536
%N Numbers whose decimal representation consist of nested and /or concatenated ordered pairs 0-9, 1-8, 2-7, 3-6 and 4-5.
%C Since A010888(a(n)) = 9 the sequence is free of primes. For squares see A065752. Replacing "0" by "(", "9" by ")", "1" by "[", "8" by "]", "2" by "{", "7" by "}", "3" by "<", "6" by ">", "4" by "d" and "5" by "b" yields well formed bracket expressions - but not the full Dyck-language, as the defining grammar is not generating words with leading zeros.
%C For example [], {}, <>, db, [()], [[]], [{}], [<>], [db], [](), [][], []{}, []<>, []db, {()}, {[]}, {{}}, {<>}, {db}, {}(), {}[], {}{}, {}<>, {}db, <()>, <[]>, <{}>, <<>>, <db>, <>(), <>[], <>{}, <><>, <>db, d()b, d[]b, d{}b, d<>b, ddbb, db(), db[], db{}, db<>, dbdb, [(())], [([])], [({})], [(<>)], [(db)], [()()], [()[]], [(){}], [()<>], [()db], [()](), [()][], [()]{}, [()]<>, [()]db, [[()]], [[[]]], [[{}]], [[<>]], [[db]], [[]()], [[][]], [[]{}], [[]<>], [[]db], [[]](), [[]][], [[]]{}, [[]]<>, [[]]db, [{()}], [{[]}], [{{}}], [{<>}], [{db}], [{}()].
%F Grammar with auxiliaries s and x (s as axiom), decimal digits as terminal alphabet and 27 rules in BNF: s : := 18 | 27 | 36 | 45 | 1x8 | 2x7 | 3x6 | 4x5 | 18x | 27x | 36x | 45x | 1x8x | 2x7x | 3x6x | 4x5x x : := 09 | 18 | 27 | 36 | 45 | 0x9 | 1x8 | 2x7 | 3x6 | 4x5 | xx.
%F Empirical g.f.: -9*(100*x^19 +x^15 +50*x^14 -10*x^13 -10*x^12 -10*x^11 -9*x^10 -156*x^9 -11*x^8 -11*x^7 -11*x^6 -12*x^5 -117*x^4 -x^3 -x^2 -x -2) / ((x -1)^2 * (x +1)^2 * (x^4 -x^3 +x^2 -x +1)^2 * (x^4 +x^3 +x^2 +x +1)). - _Colin Barker_, Mar 29 2013
%e Parsing A065752(5) = 1274275809 <- 1x4275809 <- 1x4x5809 <- 1xx809 <- 1x809 <- 1x8x <- s.
%K base,nonn
%O 0,1
%A _Reinhard Zumkeller_, Nov 17 2001