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

9, 98, 987, 9876, 98765, 987654, 9876543, 98765432, 987654321, 9876543210, 98765432099, 987654320988, 9876543209877, 98765432098766, 987654320987655, 9876543209876544, 98765432098765433, 987654320987654322, 9876543209876543211, 98765432098765432100, 987654320987654320989

Offset: 1

Author

Henry Bottomley, Sep 26 2001

Keywords

• nonn
• base
• easy

Comments

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

Examples

			Curious multiplications:
1*8 + 1 = 9;
12*8 + 2 = 98;
123*8 + 3 = 987;
1234*8 + 4 = 9876;
12345*8 + 5 = 98765;
123456*8 + 6 = 987654;
1234567*8 + 7 = 9876543;
12345678*8 + 8 = 98765432;
123456789*8 + 9 = 987654321.
- _Vincenzo Librandi_, Aug 07 2010 and _Philippe Deléham_, Mar 09 2014
		

References

• Alfred S. Posamentier, Math Charmers, Tantalizing Tidbits for the Mind, Prometheus Books, NY, 2003, page 29.

Links

• Harry J. Smith, Table of n, a(n) for n = 1..150
• Index entries for linear recurrences with constant coefficients, signature (12,-21,10).

Crossrefs

Cf. A002275, A014824, A064616.

Programs

Maple

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

Mathematica

Table[(10^n - 1)*(80/81) + n/9, {n, 20}] (* Wesley Ivan Hurt, Mar 10 2014 *)
LinearRecurrence[{12,-21,10},{9,98,987},30] (* Harvey P. Dale, Aug 20 2023 *)

PARI

{ a=0; for (n=1, 150, a=10*a + 10 - n; write("b064617.txt", n, " ", a) ) } \\ Harry J. Smith, Sep 20 2009

PARI

Vec(x*(10*x-9)/((x-1)^2*(10*x-1)) + O(x^100)) \\ Colin Barker, Sep 15 2014

Formula

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

From Colin Barker, Sep 15 2014: (Start)

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

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

E.g.f.: exp(x)*(80*exp(9*x) + 9*x - 80)/81. - Stefano Spezia, May 28 2023

Extensions

More terms from Colin Barker, Sep 15 2014