A101467 Number of distinct n-term ratios x_1 : x_2 : ... : x_n where each x_i is in the range [1-10].

10, 63, 841, 9279, 96601, 983583, 9919561, 99602559, 998026681, 9990174303, 99950992681, 999755323839, 9998777694361, 99993891685023, 999969468040201, 9999847368997119, 99999236931275641, 999996184915051743, 9999980925350886121, 99999904629080526399

Offset: 1

Author

Su Jianning (sujianning(AT)yahoo.com.cn), Jan 21 2005

Keywords

• nonn
• easy

Comments

Number of elements of {1,...,10}^n with gcd 1. - Robert Israel, Nov 28 2014

Examples

			For n=2: Consider the ratios 1:1, 1:2, ..., 1:10, 2:1, 2:2, ..., 2:10, ..., 10:1, 10:2, ..., 10:10. We get 63 different ratios from the 100 numbers list above after removing duplication. So a(2) = 63, and this is A018805(10).
		

Links

• Colin Barker, Table of n, a(n) for n = 1..999
• Index entries for linear recurrences with constant coefficients, signature (21,-151,471,-640,300).

Crossrefs

Cf. A018805 (2 terms), A071778 (3 terms), A082540 (4 terms), A082544 (5 terms).

Programs

Maple

1, seq(10^n - 5^n - 3^n - 2^n + 1, n=2..20); # Robert Israel, Nov 28 2014

PARI

Vec(x*(2700*x^5-5460*x^4+3579*x^3-1028*x^2+147*x-10)/((x-1)*(2*x-1)*(3*x-1)*(5*x-1)*(10*x-1)) + O(x^100)) \\ Colin Barker, Nov 28 2014

Formula

a(1) = 10; for n>1, a(n) = 10^n - 5^n - 3^n - 2^n + 1.

G.f.: x*(2700*x^5-5460*x^4+3579*x^3-1028*x^2+147*x-10) / ((x-1)*(2*x-1)*(3*x-1)*(5*x-1)*(10*x-1)). - Colin Barker, Nov 28 2014

a(n+4) = -300*a(n)+340*a(n+1)-131*a(n+2)+20*a(n+3)+72 for n >= 2. - Robert Israel, Dec 02 2014

a(n) = 21*a(n-1) - 151*a(n-2) + 471*a(n-3) - 640*a(n-4) + 300*a(n-5) for n > 6. - Chai Wah Wu, Apr 15 2021