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

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

Offset

0,2

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

Links

Table of n, a(n) for n=0..12.

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).

Formula

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

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

Example

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.

Mathematica

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

Crossrefs

Cf. A002275, A095371.

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

Cf. A329914, A329915.

Sequence in context: A006244 A054216 A109627 * A165154 A015261 A370780

Adjacent sequences: A095369 A095370 A095371 * A095373 A095374 A095375

Keyword

nonn,easy,base

Author

Labos Elemer, Jun 07 2004

Status

approved