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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,changed 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

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