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A047855
a(n) = T(7, n), array T given by A047848.
19
1, 2, 12, 112, 1112, 11112, 111112, 1111112, 11111112, 111111112, 1111111112, 11111111112, 111111111112, 1111111111112, 11111111111112, 111111111111112, 1111111111111112, 11111111111111112, 111111111111111112, 1111111111111111112, 11111111111111111112
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
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
n-th difference of a(n), a(n-1), ..., a(0) is 9^(n-1) for n = 1, 2, 3, ...
Cf. A164898.
Sequence in context: A102659 A212659 A191895 * A199045 A009232 A349311
KEYWORD
nonn,easy
EXTENSIONS
More terms from Harvey P. Dale, Apr 17 2013
STATUS
approved