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A064617 a(n) = (10^n - 1)*(80/81) + n/9.

%I A064617 #42 May 23 2025 11:07:42
%S A064617 9,98,987,9876,98765,987654,9876543,98765432,987654321,9876543210,
%T A064617 98765432099,987654320988,9876543209877,98765432098766,
%U A064617 987654320987655,9876543209876544,98765432098765433,987654320987654322,9876543209876543211,98765432098765432100,987654320987654320989
%N A064617 a(n) = (10^n - 1)*(80/81) + n/9.
%C A064617 a(n)/10^n converges to 80/81 = 0.987654320987654320...
%D A064617 Alfred S. Posamentier, Math Charmers, Tantalizing Tidbits for the Mind, Prometheus Books, NY, 2003, page 29.
%H A064617 Harry J. Smith, <a href="/A064617/b064617.txt">Table of n, a(n) for n = 1..150</a>
%H A064617 <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (12,-21,10).
%F A064617 a(n) = 10*a(n-1) + 10 - n = (19 - n)*A002275(n) - A064616(n) = 10*A002275(n) - A014824(n).
%F A064617 From _Colin Barker_, Sep 15 2014: (Start)
%F A064617 a(n) = 12*a(n-1) - 21*a(n-2) + 10*a(n-3).
%F A064617 G.f.: x*(10*x - 9)/((x - 1)^2*(10*x - 1)). (End)
%F A064617 E.g.f.: exp(x)*(80*exp(9*x) + 9*x - 80)/81. - _Stefano Spezia_, May 28 2023
%e A064617 Curious multiplications:
%e A064617 1*8 + 1 = 9;
%e A064617 12*8 + 2 = 98;
%e A064617 123*8 + 3 = 987;
%e A064617 1234*8 + 4 = 9876;
%e A064617 12345*8 + 5 = 98765;
%e A064617 123456*8 + 6 = 987654;
%e A064617 1234567*8 + 7 = 9876543;
%e A064617 12345678*8 + 8 = 98765432;
%e A064617 123456789*8 + 9 = 987654321.
%e A064617 - _Vincenzo Librandi_, Aug 07 2010 and _Philippe Deléham_, Mar 09 2014
%p A064617 A064617:=n->(10^n-1)*(80/81)+n/9; seq(A064617(n), 1..20); # _Wesley Ivan Hurt_, Mar 10 2014
%t A064617 Table[(10^n - 1)*(80/81) + n/9, {n, 20}] (* _Wesley Ivan Hurt_, Mar 10 2014 *)
%t A064617 LinearRecurrence[{12,-21,10},{9,98,987},30] (* _Harvey P. Dale_, Aug 20 2023 *)
%o A064617 (PARI) { a=0; for (n=1, 150, a=10*a + 10 - n; write("b064617.txt", n, " ", a) ) } \\ _Harry J. Smith_, Sep 20 2009
%o A064617 (PARI) Vec(x*(10*x-9)/((x-1)^2*(10*x-1)) + O(x^100)) \\ _Colin Barker_, Sep 15 2014
%Y A064617 Cf. A002275, A014824, A064616.
%K A064617 nonn,base,easy
%O A064617 1,1
%A A064617 _Henry Bottomley_, Sep 26 2001
%E A064617 More terms from _Colin Barker_, Sep 15 2014