A212704 a(n) = 9*n*10^(n-1).

9, 180, 2700, 36000, 450000, 5400000, 63000000, 720000000, 8100000000, 90000000000, 990000000000, 10800000000000, 117000000000000, 1260000000000000, 13500000000000000, 144000000000000000, 1530000000000000000, 16200000000000000000, 171000000000000000000, 1800000000000000000000

Offset: 1

Author

Stanislav Sykora, May 25 2012

Keywords

• nonn
• easy

Comments

Main transitions in systems of n particles with spin 9/2.

Please, refer to the general explanation in A212697.

This particular sequence is obtained for base b=10, corresponding to spin S = (b-1)/2 = 9/2.

Number of 0 needed to write all numbers of n+1 digits. - Bruno Berselli, Jun 30 2014

Essentially the same as A113119. - Bernard Schott, Nov 15 2022

From Bernard Schott, Nov 22 2022: (Start)

Number of nonzero digits needed to write all integers from 1 up to 10^n - 1.

a(n) is a square iff n in { A016754 union A033583\{0} } (see formulas). (End)

Links

• Stanislav Sykora, Table of n, a(n) for n = 1..100
• Stanislav Sýkora, Magnetic Resonance on OEIS, Stan's NMR Blog (Dec 31, 2014), Retrieved Nov 12, 2019.
• Index entries for linear recurrences with constant coefficients, signature (20,-100).

Crossrefs

Cf. A008591, A011557, A016754, A033583, A033713, A033714, A053541, A081045, A113119.

Cf. A001787, A212697, A212698, A212699, A212700, A212701, A212702, A212703 (for b=2..9).

Programs

Mathematica

Rest@ CoefficientList[Series[9 x/(10 x - 1)^2, {x, 0, 18}], x] (* or *)
Array[9 # 10^(# - 1) &, 18] (* Michael De Vlieger, Nov 18 2019 *)

PARI

mtrans(n, b) = n*(b-1)*b^(n-1);
a(n) = mtrans(n, 10);

Python

def a(n): return 9*n*10**(n-1)
print([a(n) for n in range(1, 21)]) # Michael S. Branicky, Nov 14 2022

Formula

a(n) = n*(b-1)*b^(n-1) with b=10.

From R. J. Mathar, Oct 15 2013: (Start)

G.f.: 9*x/(10*x-1)^2.

a(n) = 9*A053541(n). (End)

From Bernard Schott, Nov 14 2022: (Start)

a(n+1) - a(n) = 9*A081045(n).

a(n) = A113119(n) for n > 1.

a(n) = A033713(n+1) - A033713(n) = A033714(n+1) - A033714(n).

a(A016754(n)) = (3 * (2n+1) * 10^(2*n*(n+1)))^2.

a(A033583(n)) = (3 * n * 10^(5*n^2))^2. (End)

From Elmo R. Oliveira, May 13 2025: (Start)

E.g.f.: 9*x*exp(10*x).

a(n) = A008591(n)*A011557(n-1).

a(n) = 20*a(n-1) - 100*a(n-2) for n > 2. (End)