A014925 Number of zeros in numbers 1 to 111...1 (n+1 digits).
1, 21, 321, 4321, 54321, 654321, 7654321, 87654321, 987654321, 10987654321, 120987654321, 1320987654321, 14320987654321, 154320987654321, 1654320987654321, 17654320987654321, 187654320987654321, 1987654320987654321, 20987654320987654321, 220987654320987654321
Offset: 1
Links
- Index entries for linear recurrences with constant coefficients, signature (21,-120,100).
Crossrefs
Cf. A033713.
Programs
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Magma
[(10^n*(9*n-1)+1)/81: n in [1..25]]; // Vincenzo Librandi, Mar 31 2011
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PARI
a(n) = sum(i=0, n, i*10^(i-1)); \\ Michel Marcus, Dec 15 2016
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PARI
my(x='x+O('x^21)); Vec(-x/((x-1)*(10*x-1)^2)) \\ Elmo R. Oliveira, May 24 2025
Formula
a(1) = 1; for n>1, a(n) = n*10^(n-1) + a(n-1).
G.f.: x/((1-x)*(1-10*x)^2).
a(n) = 10^n*(n+1)/9 - (1/81)*10^(n+1) + 1/81.
a(n) = (10^n*(9*n - 1) + 1)/81. - Kenneth E. Caviness, Mar 30 2011
E.g.f.: (1 - exp(9*x) + 90*x*exp(9*x))*exp(x)/81. - Ilya Gutkovskiy, May 02 2016
a(n) = Sum_{i=0..n} i*10^(i-1). - José de Jesús Camacho Medina, Dec 14 2016
From Elmo R. Oliveira, May 24 2025: (Start)
a(n) = 20*a(n-1) - 100*a(n-2) + 1 for n > 2.
a(n) = 21*a(n-1) - 120*a(n-2) + 100*a(n-3) for n > 3. (End)
Extensions
Better description from Stephen G Penrice, Oct 03 2000
More terms from Elmo R. Oliveira, May 24 2025