OFFSET
0,2
COMMENTS
Range of A164898, apart from first term. - Reinhard Zumkeller, Aug 30 2009
a(n) is the number of integers less than or equal to 10^n, whose initial digit is 1. - Michel Marcus, Jul 04 2019
a(n) is 2^n represented in bijective base-2 numeration. - Alois P. Heinz, Aug 26 2019
This sequence proves both A028842 (numbers with prime product of digits) and A028843 (numbers with prime iterated product of digits) are infinite. Proof: Suppose either of those sequences is finite. Label as omega the supposed last term. Compute n = ceiling(log_10 omega) + 1. Then a(n) > omega. The product of digits of a(n) is 2, contradicting the assumption that omega is the final term of either A028842 or A028843. - Alonso del Arte, Apr 14 2020
For n >= 2, the concatenation of a(n) with 8*a(n) equals (3*R_n+3)^2, where R_n = A002275(n) is the repunit with n 1's; hence this sequence, except for {1,2}, is a subsequence of A115549. - Bernard Schott, Apr 30 2022
LINKS
Ivan Panchenko, Table of n, a(n) for n = 0..200
Wikipedia, Bijective numeration
Index entries for linear recurrences with constant coefficients, signature (11,-10).
FORMULA
a(n) = (10^n + 8)/9. - Ralf Stephan, Feb 14 2004
a(0) = 1, a(1) = 2, a(n) = 11*a(n - 1) - 10*a(n - 2) for n > 1. - Lambert Klasen (lambert.klasen(AT)gmx.net), Jan 28 2005
G.f.: (1 - 9*x)/(1 - 11*x + 10*x^2). - Philippe Deléham, Oct 05 2009
a(n) = 10*a(n-1) - 8 (with a(0) = 1). - Vincenzo Librandi, Aug 06 2010
MAPLE
a[0]:=0:a[1]:=1:for n from 2 to 50 do a[n]:=10*a[n-1]+1 od: seq(a[n]+1, n=0..18); # Zerinvary Lajos, Mar 20 2008
MATHEMATICA
Join[{1}, Table[FromDigits[PadLeft[{2}, n, 1]], {n, 30}]] (* Harvey P. Dale, Apr 17 2013 *)
(10^Range[0, 29] + 8)/9 (* Alonso del Arte, Apr 12 2020 *)
PROG
(PARI) a(n)=if(n==0, 1, if(n==1, 2, 11*a(n-1)-10*a(n-2)))
for(i=0, 10, print1(a(i), ", ")) \\ Lambert Klasen, Jan 28 2005
(Sage) [gaussian_binomial(n, 1, 10)+1 for n in range(17)] # Zerinvary Lajos, May 29 2009
(Scala) (List.fill(20)(10: BigInt)).scanLeft(1: BigInt)(_ * _).map(n => (n + 8)/9) // Alonso del Arte, Apr 12 2020
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
EXTENSIONS
More terms from Harvey P. Dale, Apr 17 2013
STATUS
approved