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A060531
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9th binomial transform of (1,0,1,0,1,...), A059841.
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8
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1, 9, 82, 756, 7048, 66384, 631072, 6048576, 58388608, 567108864, 5536870912, 54294967296, 534359738368, 5274877906944, 52199023255552, 517592186044416, 5140737488355328, 51125899906842624, 509007199254740992, 5072057594037927936, 50576460752303423488
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OFFSET
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0,2
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COMMENTS
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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
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LINKS
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Harry J. Smith, Table of n, a(n) for n = 0..200
Index entries for linear recurrences with constant coefficients, signature (18,-80).
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FORMULA
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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.
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EXAMPLE
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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}.
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MAPLE
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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:
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MATHEMATICA
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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 *)
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PROG
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(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
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CROSSREFS
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Cf. A007582, A081192.
Sequence in context: A068109 A163460 A081191 * A248848 A045741 A283498
Adjacent sequences: A060528 A060529 A060530 * A060532 A060533 A060534
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KEYWORD
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nonn,easy
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AUTHOR
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N. J. A. Sloane, Apr 12 2001
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EXTENSIONS
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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
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STATUS
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approved
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