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A060531 9th binomial transform of (1,0,1,0,1,...), A059841. 8
1, 9, 82, 756, 7048, 66384, 631072, 6048576, 58388608, 567108864, 5536870912, 54294967296, 534359738368, 5274877906944, 52199023255552, 517592186044416, 5140737488355328, 51125899906842624, 509007199254740992, 5072057594037927936, 50576460752303423488 (list; graph; refs; listen; history; text; internal format)
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

A060531 := proc(n) option remember: if n = 1 then RETURN(9) fi: 8*A060531(n-1) + 10^(n-1): end: for n from 1 to 40 do printf(`%d, `, A060531(n)) od:

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

Cf. A007582, A081192.

Sequence in context: A068109 A163460 A081191 * A248848 A045741 A110567

Adjacent sequences:  A060528 A060529 A060530 * A060532 A060533 A060534

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 and new name by Wolfdieter Lang, Jul 18 2017

STATUS

approved

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Last modified September 20 16:05 EDT 2017. Contains 292276 sequences.