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a(n) = (10^n - 1)*(80/81) + n/9.
9

%I #38 Aug 20 2023 12:40:29

%S 9,98,987,9876,98765,987654,9876543,98765432,987654321,9876543210,

%T 98765432099,987654320988,9876543209877,98765432098766,

%U 987654320987655,9876543209876544,98765432098765433,987654320987654322,9876543209876543211,98765432098765432100,987654320987654320989

%N a(n) = (10^n - 1)*(80/81) + n/9.

%C a(n)/10^n converges to 80/81 = 0.987654320987654320...

%H Harry J. Smith, <a href="/A064617/b064617.txt">Table of n, a(n) for n = 1..150</a>

%H <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (12,-21,10).

%F a(n) = 10*a(n-1) + 10 - n = (19 - n)*A002275(n) - A064616(n) = 10*A002275(n) - A014824(n).

%F From _Colin Barker_, Sep 15 2014: (Start)

%F a(n) = 12*a(n-1) - 21*a(n-2) + 10*a(n-3).

%F G.f.: x*(10*x - 9)/((x - 1)^2*(10*x - 1)). (End)

%F E.g.f.: exp(x)*(80*exp(9*x) + 9*x - 80)/81. - _Stefano Spezia_, May 28 2023

%e Curious multiplications:

%e 1*8 + 1 = 9;

%e 12*8 + 2 = 98;

%e 123*8 + 3 = 987;

%e 1234*8 + 4 = 9876;

%e 12345*8 + 5 = 98765;

%e 123456*8 + 6 = 987654;

%e 1234567*8 + 7 = 9876543;

%e 12345678*8 + 8 = 98765432;

%e 123456789*8 + 9 = 987654321.

%e - _Vincenzo Librandi_, Aug 07 2010 and _Philippe Deléham_, Mar 09 2014

%p A064617:=n->(10^n-1)*(80/81)+n/9; seq(A064617(n), 1..20); # _Wesley Ivan Hurt_, Mar 10 2014

%t Table[(10^n - 1)*(80/81) + n/9, {n, 20}] (* _Wesley Ivan Hurt_, Mar 10 2014 *)

%t LinearRecurrence[{12,-21,10},{9,98,987},30] (* _Harvey P. Dale_, Aug 20 2023 *)

%o (PARI) { a=0; for (n=1, 150, a=10*a + 10 - n; write("b064617.txt", n, " ", a) ) } \\ _Harry J. Smith_, Sep 20 2009

%o (PARI) Vec(x*(10*x-9)/((x-1)^2*(10*x-1)) + O(x^100)) \\ _Colin Barker_, Sep 15 2014

%Y Cf. A002275, A014824, A064616.

%K nonn,base,easy

%O 1,1

%A _Henry Bottomley_, Sep 26 2001

%E More terms from _Colin Barker_, Sep 15 2014