A109242 Expansion of 1/((1-x)(1-10x)(1-100x)).

1, 111, 11211, 1122211, 112232211, 11223332211, 1122334332211, 112233444332211, 11223344544332211, 1122334455544332211, 112233445565544332211, 11223344556665544332211, 1122334455667665544332211, 112233445566777665544332211, 11223344556677877665544332211

Offset: 0

Author

Paul Barry, Jun 23 2005

Keywords

• easy
• nonn

Comments

Partial sums of A109241.

Examples

			The numbers of 1's, 2's, 3's etc. appearing occur according to
1:1,3,4,4,4,4,4,4,...
2:0,0,1,3,4,4,4,4,...
3:0,0,0,0,1,3,4,4,...
4:0,0,0,0,0,0,1,3,... etc. up to term 17, where 9->10 etc. changes the pattern.
		

Links

• Andrew Howroyd, Table of n, a(n) for n = 0..100
• Index entries for linear recurrences with constant coefficients, signature (111,-1110,1000).

Programs

PARI

a(n) = {10^(2*n+3)/891 - 10^(n+1)/81 + 1/891} \\ Andrew Howroyd, Nov 08 2019

Sage

[gaussian_binomial(n,2,10) for n in range(2,14)] # Zerinvary Lajos, May 27 2009

Formula

a(n) = 10^(2*n+3)/891 - 10^(n+1)/81 + 1/891.

a(n) = 111*a(n-1) - 1110*a(n-2) + 1000*a(n-3).

a(n) = 110*a(n-1) - 1000*a(n-2) + 1, n >= 2. - Vincenzo Librandi, Mar 18 2011

Extensions

Terms a(12) and beyond from Andrew Howroyd, Nov 08 2019