OFFSET
1,2
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.
LINKS
Vincenzo Librandi, Table of n, a(n) for n = 1..200
Index entries for linear recurrences with constant coefficients, signature (11,-10)
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
EXAMPLE
T(19) = 190 begins with 19. Hence 19 is a term of the sequence.
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 *)
PROG
(Magma) [2*10^(n-1)-1 : n in [1..20]]; // Vincenzo Librandi, Nov 01 2011
(PARI) a(n) = 2*10^(n-1)-1; \\ Michel Marcus, Jul 06 2024
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Joseph L. Pe, Feb 21 2002
EXTENSIONS
a(7)-a(19) from Vincenzo Librandi, Nov 01 2011
STATUS
approved