A093299 a(n) is the largest number such that all of a(n)'s length-n substrings are distinct and divisible by 99.

0, 99, 990, 9900990, 99990, 99990099990, 9999990, 999999009999990, 999999990, 9999999900999999990, 99999999990, 99999999990099999999990, 9999999999990, 999999999999009999999999990, 999999999999990, 9999999999999900999999999999990, 99999999999999990, 99999999999999990099999999999999990

Offset

1,2

Comments

From Luca Zanardi Lamardo, Jan 26 2025: (Start)

The formulas for a(n) can be found by analyzing numbers (mod 99) and noticing two properties:

If n is odd, the number can only have n digits.

If n is even, any 2 digits that have n-1 digits between them must be equal. (End)

Links

Luca Zanardi Lamardo, Table of n, a(n) for n = 1..500

Hans Havermann, Table of A093211-A093299

Index entries for linear recurrences with constant coefficients, signature (1,10100,-10100,-1000000,1000000).

Formula

From Luca Zanardi Lamardo, Jan 26 2025: (Start)

a(2k+1) = 10^(2k+1) - 10, being digits 99...990 with 2k 9's.

a(2k) = 10^(4k-1)-10^(2k+1) + 10^(2k-1) - 10 for k>=2, being digits 99...990099...990 with 2k-2 9's in each of the groups of 9's. (End)

Example

a(4) is 9900990 because its length-4 substrings (9900, 9009, 0099, 0990) are all distinct and divisible by 99 and there is no larger number with this property.

Prog

(C++)
#include <string>
using namespace std;
string a093299(int n) {
    if(n == 1) return "0";
    else if(n == 2) return "99";
    else {
        if(n % 2 == 1) return string(n - 1, '9') + "0";
        else return string(n - 2, '9') + "00" + string(n - 2, '9') + "0";
    }
}

Crossrefs

Cf. A093211, A093212, ..., A093298.

Sequence in context: A223157 A223155 A308466 * A196645 A196600 A074786

Adjacent sequences: A093296 A093297 A093298 * A093300 A093301 A093302

Keyword

nonn,base,easy,changed

Author

Hans Havermann, Mar 29 2004

Extensions

a(9)-a(13) from John Cerkan, Jun 14 2017

a(14)-a(18) from Luca Zanardi Lamardo, Jan 26 2025

Status

approved