OFFSET
0,4
LINKS
Alois P. Heinz, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (20,-100,-9,99,-90).
FORMULA
a(0) = a(1) = 0, a(2) = 1, a(n) = 9*(10^(n-3) - a(n-3) + Sum_{i=2..n-1} a(i)) for n>=3.
G.f.: x^2*(x-1)^2/((10*x-1)*(9*x^4-9*x^3+10*x-1)). - Alois P. Heinz, Feb 26 2015
EXAMPLE
a(2) = 1 because there is only 1 two-digit string that contains the substring "00", i.e., "00" itself.
a(3) = 18 because there are 18 three-digit strings that contain a "00" substring that is not part of a string of three or more consecutive "0" digits; using "+" to represent a nonzero digit, the 18 strings comprise 9 of the form "00+" and 9 of the form "+00". ("000" is excluded.)
a(4) = 261 because there are 261 four-digit strings that contain a "00" substring that is not part of a string of three or more consecutive "0" digits; using "+" as above and "." to denote any digit (0 or otherwise), the 261 strings comprise 9*10=90 of the form "00+.", 9*9=81 of the form "+00+", and 10*9=90 of the form ".+00".
a(5) = 3411 because there are 3411 five-digit strings that contain at least one "00" substring that is not part of a string of three or more consecutive "0" digits; using "+" and "." as above, the 3411 strings comprise 9*10*10=900 of the form "00+..", 9*9*10=810 of the form "+00+.", 10*9*9=810 of the form ".+00+", and 99*9=891 that are of the form "..+00" but not of the form "00+00" (since the 9 strings of that latter form were already counted among the 900 of the form "00+..").
MATHEMATICA
LinearRecurrence[{20, -100, -9, 99, -90}, {0, 0, 1, 18, 261}, 30] (* Harvey P. Dale, Jan 01 2021 *)
CROSSREFS
KEYWORD
nonn,base,easy
AUTHOR
Jon E. Schoenfield, Feb 21 2015
STATUS
approved