OFFSET

0,1

COMMENTS

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.

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}], [{}()].

FORMULA

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.

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

EXAMPLE

Parsing A065752(5) = 1274275809 <- 1x4275809 <- 1x4x5809 <- 1xx809 <- 1x809 <- 1x8x <- s.

CROSSREFS

KEYWORD

base,nonn

AUTHOR

Reinhard Zumkeller, Nov 17 2001

STATUS

approved