A060531 9th binomial transform of (1,0,1,0,1,...), A059841.

1, 9, 82, 756, 7048, 66384, 631072, 6048576, 58388608, 567108864, 5536870912, 54294967296, 534359738368, 5274877906944, 52199023255552, 517592186044416, 5140737488355328, 51125899906842624, 509007199254740992, 5072057594037927936, 50576460752303423488

Offset: 0

Author

N. J. A. Sloane, Apr 12 2001

Keywords

• nonn
• easy

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

Examples

			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}.
		

Links

• Harry J. Smith, Table of n, a(n) for n = 0..200
• Index entries for linear recurrences with constant coefficients, signature (18,-80).

Crossrefs

Cf. A007582, A081192.

Programs

Magma

[(8^n+10^n)/2: n in [0..20]]; // Vincenzo Librandi, Jul 18 2017

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 *)

PARI

a(n) = { (8^n + 10^n)/2 } \\ Harry J. Smith, Jul 06 2009

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.

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