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

%I A060531 #48 Dec 29 2024 14:24:48
%S A060531 1,9,82,756,7048,66384,631072,6048576,58388608,567108864,5536870912,
%T A060531 54294967296,534359738368,5274877906944,52199023255552,
%U A060531 517592186044416,5140737488355328,51125899906842624,509007199254740992,5072057594037927936,50576460752303423488
%N A060531 9th binomial transform of (1,0,1,0,1,...), A059841.
%C A060531 Binomial transform of A081190.
%C A060531 Number of strings of length n of the decimal digits 0..9 that contain an even number of 0's.
%C A060531 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
%H A060531 Harry J. Smith, <a href="/A060531/b060531.txt">Table of n, a(n) for n = 0..200</a>
%H A060531 <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (18,-80).
%F A060531 G.f.: (1 - 9*x)/((1 - 8*x)*(1 -10*x)).
%F A060531 E.g.f.: exp(9*x)*cosh(x).
%F A060531 a(n) = (8^n + 10^n)/2 = 2^(n-1)*(4^n + 5^n).
%F A060531 a(n) = 18*a(n-1) - 80*a(n-2), a(0) = 1, a(1) = 9.
%F A060531 a(n) = 8*a(n-1) + 10^(n-1), a(1) = 9.
%e A060531 For n = 1 there are 9 strings: {1 2 3 4 5 6 7 8 9};
%e A060531 for n = 2 there are 82: {00 11 12 13 14 15 16 17 18 19 21 ... 96 97 98 99}.
%p A060531 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:
%t A060531 Table[8^n/2 + 10^n/2, {n, 0, 19}] (* or *)
%t A060531 LinearRecurrence[{18, -80}, {1, 9}, 19] (* or *)
%t A060531 CoefficientList[Series[(1 - 9 x)/((1 - 8 x) (1 - 10 x)), {x, 0, 19}], x] (* _Michael De Vlieger_, Jul 17 2017 *)
%o A060531 (PARI) a(n) = { (8^n + 10^n)/2 } \\ _Harry J. Smith_, Jul 06 2009
%o A060531 (Magma) [(8^n+10^n)/2: n in [0..20]]; // _Vincenzo Librandi_, Jul 18 2017
%Y A060531 Cf. A007582, A081192.
%K A060531 nonn,easy
%O A060531 0,2
%A A060531 _N. J. A. Sloane_, Apr 12 2001
%E A060531 Additional comments from _Paul Barry_, Mar 11 2003
%E A060531 Typo in definition corrected by _Paolo P. Lava_, Sep 18 2008
%E A060531 Edited by and new name from _Wolfdieter Lang_, Jul 18 2017