A067272 a(n) = 2*10^(n-1) - 1.

1, 19, 199, 1999, 19999, 199999, 1999999, 19999999, 199999999, 1999999999, 19999999999, 199999999999, 1999999999999, 19999999999999, 199999999999999, 1999999999999999, 19999999999999999, 199999999999999999, 1999999999999999999, 19999999999999999999, 199999999999999999999

Offset: 1

Author

Joseph L. Pe, Feb 21 2002

Keywords

• nonn
• easy

Comments

Smaller of the smallest pair of successive n-digit numbers which have no digit in common: (1,2), (19,20), (199,200) etc. - Amarnath Murthy, Nov 10 2002

Original name: Numbers n such that the digits of T(n) = n*(n+1)/2, the n-th triangular number, begin with n.

Examples

			T(19) = 190 begins with 19. Hence 19 is a term of the sequence.
		

Links

• Vincenzo Librandi, Table of n, a(n) for n = 1..200
• Index entries for linear recurrences with constant coefficients, signature (11,-10).

Programs

Magma

[2*10^(n-1)-1 : n in [1..20]]; // Vincenzo Librandi, Nov 01 2011

Mathematica

(*returns true if a begins with b, false o.w.*) f2[a_, b_] := Module[{c, d, e, g, h, i, r}, r = False; c = ToString[a]; d = ToString[b]; g = StringPosition[c, d]; h = Length[g]; If[h > 0, i = g[[h]]; If[i[[1]] == 1, r = True]]; r]; Do[If[f2[n(n + 1)/2, n], Print[n]], {n, 1, 10^6} ]
CoefficientList[Series[(1 + 8 x)/((1 - x) (1 - 10 x)), {x, 0, 20}], x] (* Vincenzo Librandi, Aug 13 2014 *)

PARI

a(n) = 2*10^(n-1)-1; \\ Michel Marcus, Jul 06 2024

Formula

a(n) = 2*10^(n-1) - 1. - Benoit Cloitre, Feb 28 2002

a(n) = 10*a(n-1) + 9. - Vincenzo Librandi, Nov 01 2011

G.f.: x*(1+8*x)/((1-x)*(1-10*x)). - Vincenzo Librandi, Aug 13 2014

From Elmo R. Oliveira, Jun 14 2025: (Start)

E.g.f.: (4 - 5*exp(x) + exp(10*x))/5.

a(n) = 11*a(n-1) - 10*a(n-2) for n > 2. (End)

Extensions

a(7)-a(19) from Vincenzo Librandi, Nov 01 2011