OFFSET
0,2
COMMENTS
Binomial transform of A081190.
Number of strings of length n of the decimal digits 0..9 that contain an even number of 0's.
An equivalent formulation is: a(n) is also the number of words of length n over an alphabet of ten letters with a chosen letter appearing an even number of times. See a comment in A007582, also for the cross references for the 1- to 11-letter word cases. - Wolfdieter Lang, Jul 17 2017
LINKS
Harry J. Smith, Table of n, a(n) for n = 0..200
Index entries for linear recurrences with constant coefficients, signature (18,-80).
FORMULA
G.f.: (1 - 9*x)/((1 - 8*x)*(1 -10*x)).
E.g.f.: exp(9*x)*cosh(x).
a(n) = (8^n + 10^n)/2 = 2^(n-1)*(4^n + 5^n).
a(n) = 18*a(n-1) - 80*a(n-2), a(0) = 1, a(1) = 9.
a(n) = 8*a(n-1) + 10^(n-1), a(1) = 9.
EXAMPLE
For n = 1 there are 9 strings: {1 2 3 4 5 6 7 8 9};
for n = 2 there are 82: {00 11 12 13 14 15 16 17 18 19 21 ... 96 97 98 99}.
MAPLE
MATHEMATICA
Table[8^n/2 + 10^n/2, {n, 0, 19}] (* or *)
LinearRecurrence[{18, -80}, {1, 9}, 19] (* or *)
CoefficientList[Series[(1 - 9 x)/((1 - 8 x) (1 - 10 x)), {x, 0, 19}], x] (* Michael De Vlieger, Jul 17 2017 *)
PROG
(PARI) { for (n=0, 200, if (n==0, a=1, a=8*a + 10^(n - 1)); write("b060531.txt", n, " ", a); ) } \\ Harry J. Smith, Jul 06 2009
(Magma) [(8^n+10^n)/2: n in [0..20]]; // Vincenzo Librandi, Jul 18 2017
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
N. J. A. Sloane, Apr 12 2001
EXTENSIONS
Additional comments from Paul Barry, Mar 11 2003
Typo in definition corrected by Paolo P. Lava, Sep 18 2008
Edited by and new name from Wolfdieter Lang, Jul 18 2017
STATUS
approved