%I #38 Feb 16 2024 10:15:38

%S 1,91,9091,909091,90909091,9090909091,909090909091,90909090909091,

%T 9090909090909091,909090909090909091,90909090909090909091,

%U 9090909090909090909091,909090909090909090909091

%N 1+integers repeating "90" decimal digit pattern:.

%C These numbers arise for example as divisors of several repunits (A002275).

%C The aerated sequence A(n) = [1, 0, 91, 0, 9091, 0, 909091,...] is a divisibility sequence, i.e., A(n) divides A(m) whenever n divides m. It is the case P1 = 0, P2 = -11^2, Q = 10 of the 3-parameter family of 4th-order linear divisibility sequences found by Williams and Guy. - _Peter Bala_, Aug 22 2019

%C Except for a(0) = 1, these terms M are such that 21 * M = 1M1, where 1M1 denotes the concatenation of 1, M and 1. Actually 21 is A329914(1) and a(1) = A329915(1) = 91, and the terms >=91 form the set {M_21}; for example, 21 * 909091 = 1(909091)1. - _Bernard Schott_, Dec 01 2019

%H H. C. Williams and R. K. Guy, <a href="http://dx.doi.org/10.1142/S1793042111004587">Some fourth-order linear divisibility sequences</a>, Intl. J. Number Theory 7 (5) (2011) 1255-1277.

%H H. C. Williams and R. K. Guy, <a href="https://www.emis.de/journals/INTEGERS/papers/a17self/a17self.Abstract.html">Some Monoapparitic Fourth Order Linear Divisibility Sequences</a> Integers, Volume 12A (2012) The John Selfridge Memorial Volume.

%H <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (101,-100).

%F a(n) = 1+90*(-1+100^n)/99 = (10^(2n+1)+1)/11. - _Rick L. Shepherd_, Aug 01 2004

%F a(n) = 101*a(n-1)-100*a(n-2). G.f.: -(10*x-1) / ((x-1)*(100*x-1)). - _Colin Barker_, Jul 03 2013

%e Digit-pattern P=[ab..z] repeating integers equal formally with P*(-1+10^(Ln))/(-1+10^L), where L is the length of pattern;

%e a(9) divides A002275(38) repunit. See A095371.

%t Table[1+90*(100^n-1)/99, {n, 0, 20}]

%Y Cf. A002275, A095371.

%Y Cf. A015585, A097209, A001562, A054416, A152577, A100047.

%Y Cf. A329914, A329915.

%K nonn,easy,base

%O 0,2

%A _Labos Elemer_, Jun 07 2004