A155671 -id:A155671 - cp's OEIS frontend

A016189 a(n) = 10^n - 9^n.

0, 1, 19, 271, 3439, 40951, 468559, 5217031, 56953279, 612579511, 6513215599, 68618940391, 717570463519, 7458134171671, 77123207545039, 794108867905351, 8146979811148159, 83322818300333431, 849905364703000879, 8649148282327007911, 87842334540943071199, 890581010868487640791

Offset: 0

N. J. A. Sloane

Almost all numbers contain any given sequence of digits (in any base) [Theorem 143 of Hardy and Wright]. a(7) = 5217031, more than 52% of the numbers < 10^7 contain any given nonzero decimal digit. - Frank Ellermann, May 30 2001
a(n) gives the number of integers from 0 to 10^n-1 which contain (at least) any one given decimal digit except 0. - Michael Taktikos, Aug 24 2004
These are the numerators of a(n)=(integral{x=0 to 0.2} (1-0.5*x)^n dx). E.g., a(3)=3439/20000. The denominators are b(n)=5*(n+1)*10^n. E.g., b(3)=20000. - Al Hakanson (hawkuu(AT)excite.com), Feb 22 2004
Binomial transforms of sequences defined by a(n)=(C+1)^n-C^n are the sequences (C+2)^n-(C+1)^n. The binomial transform of this here is in A016195, for example. - R. J. Mathar, Nov 27 2008
First differences are given in A088924. - M. F. Hasler, May 04 2015

G. H. Hardy and E. M. Wright, An Introduction to the Theory of Numbers, 5th ed., Oxford Univ. Press, 1979, th. 143

Vincenzo Librandi, Table of n, a(n) for n = 0..130
Alexander Bogomolny, Almost every integer has a digit 3 in it
John Elias, Illustration of Initial Terms
James Grime, 3 is everywhere, Numberphile video
Index entries for linear recurrences with constant coefficients, signature (19, -90).

Base 2: A000225, 3: A001047, 4: A005061, 5: A005060, 6: A005062, base 7: A016169, 8: A016177, 9: A016185 11: A016195 12: A016197.
Equals A155671 - 1.
Cf. A011557, A011533.

a016189 n = 10 ^ n - 9 ^ n
a016189_list = 0 : zipWith (+) (map (* 9) a016189_list) a011557_list
-- Reinhard Zumkeller, Apr 03 2015

[10^n - 9^n: n in [0..20]]; // Vincenzo Librandi, Apr 26 2011

f[n_]:=10^n-9^n;f[Range[0,40]] (* Vladimir Joseph Stephan Orlovsky, Feb 14 2011 *)

a(n)=10^n-9^n \\ M. F. Hasler, May 04 2015

G.f.: x/((1-9x)(1-10x)).
a(0) = 0, a(1) = 1, then a(n+1) = 9*a(n) + 10^n.
a(n) = 19*a(n-1) - 90*a(n-2), n > 1; a(0)=0, a(1)=1. - Philippe Deléham, Jan 01 2009
E.g.f.: e^(10*x) - e^(9*x). - Mohammad K. Azarian, Jan 14 2009

A155672 11^n-9^n+1^n.

Original entry on oeis.org

1, 3, 41, 603, 8081, 102003, 1240121, 14704203, 171312161, 1970527203, 22450640201, 253930611003, 2855998840241, 31980846315603, 356873041128281, 3971357037321003, 44096709674720321, 488769846799627203

Offset: 0

Mohammad K. Azarian, Feb 01 2009

nonn

Index entries for linear recurrences with constant coefficients, signature (21,-119,99).

A155654, A155655, A155656, A155657, A155658, A155659, A155660, A155661, A155662, A155663, A155664, A155665, A155666, A155667, A155668, A155669, A155670, A155671

Table[11^n-9^n+1,{n,0,30}] (* or *) LinearRecurrence[{21,-119,99},{1,3,41},30] (* Harvey P. Dale, May 07 2019 *)

G.f.: 1/(1-11*x)-1/(1-9*x)+1/(1-x). E.g.f.: e^(11*x)-e^(9*x)+e^x.

a(n)=20*a(n-1)-99*a(n-2)+80 with a(0)=1, a(1)=3 [From Vincenzo Librandi, Jul 21 2010]

A155673 a(n) = 11^n + 10^n - 1.

Original entry on oeis.org

1, 20, 220, 2330, 24640, 261050, 2771560, 29487170, 314358880, 3357947690, 35937424600, 385311670610, 4138428376720, 44522712143930, 479749833583240, 5177248169415650, 55949729863572160, 605447028499293770

Offset: 0

Mohammad K. Azarian, Feb 01 2009

Index entries for linear recurrences with constant coefficients, signature (22,-131,110).

Cf. A155654, A155655, A155656, A155657, A155658, A155659, A155660, A155661, A155662, A155663, A155664, A155665, A155666, A155667, A155668, A155669, A155670, A155671, A155672.

Table[11^n+10^n-1,{n,0,20}] (* Harvey P. Dale, Feb 24 2011 *)

G.f.: 1/(1-11*x)+1/(1-10*x)-1/(1-x).
E.g.f.: e^(11*x)+e^(10*x)-e^x.
a(n) = 21*a(n-1)-110*a(n-2)-90 with a(0)=1, a(1)=20. - Vincenzo Librandi, Jul 21 2010
a(n) = 22*a(n-1)-131*a(n-2)+110*a(n-3). - Wesley Ivan Hurt, May 19 2024

A155674 11^n-10^n+1^n.

Original entry on oeis.org

1, 2, 22, 332, 4642, 61052, 771562, 9487172, 114358882, 1357947692, 15937424602, 185311670612, 2138428376722, 24522712143932, 279749833583242, 3177248169415652, 35949729863572162, 405447028499293772

Offset: 0

Mohammad K. Azarian, Feb 01 2009

nonn

Index entries for linear recurrences with constant coefficients, signature (22,-131,110).

A155654, A155655, A155656, A155657, A155658, A155659, A155660, A155661, A155662, A155663, A155664, A155665, A155666, A155667, A155668, A155669, A155670, A155671, A155672, A155673

G.f.: 1/(1-11*x)-1/(1-10*x)+1/(1-x). E.g.f.: e^(11*x)-e^(10*x)+e^x.

a(n)=21*a(n-1)-110*a(n-2)+90 with a(0)=1, a(1)=2 [From Vincenzo Librandi, Jul 21 2010]

cp's OEIS Frontend

A016189 a(n) = 10^n - 9^n.

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A155672 11^n-9^n+1^n.

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A155673 a(n) = 11^n + 10^n - 1.

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A155674 11^n-10^n+1^n.

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