A095372 1+integers repeating "90" decimal digit pattern.

1, 91, 9091, 909091, 90909091, 9090909091, 909090909091, 90909090909091, 9090909090909091, 909090909090909091, 90909090909090909091, 9090909090909090909091, 909090909090909090909091

Offset: 0

Author

Labos Elemer, Jun 07 2004

Keywords

• nonn
• easy
• base

Comments

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

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

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

Examples

			Digit-pattern P=[ab..z] repeating integers equal formally with P*(-1+10^(Ln))/(-1+10^L), where L is the length of pattern;
a(9) divides A002275(38) repunit. See A095371.
		

Links

• H. C. Williams and R. K. Guy, Some fourth-order linear divisibility sequences, Intl. J. Number Theory 7 (5) (2011) 1255-1277.
• H. C. Williams and R. K. Guy, Some Monoapparitic Fourth Order Linear Divisibility Sequences Integers, Volume 12A (2012) The John Selfridge Memorial Volume.
• Index entries for linear recurrences with constant coefficients, signature (101,-100).

Crossrefs

Cf. A002275, A095371.

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

Cf. A329914, A329915.

Programs

Mathematica

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

Formula

a(n) = 1 + 90*(-1 + 100^n)/99 = (10^(2*n+1) + 1)/11. - Rick L. Shepherd, Aug 01 2004

From Colin Barker, Jul 03 2013: (Start)

a(n) = 101*a(n-1) - 100*a(n-2).

G.f.: -(10*x-1)/((x-1)*(100*x-1)). (End)

E.g.f.: exp(x)*(1 + 10*(exp(99*x) - 1)/11). - Elmo R. Oliveira, Mar 15 2025