A135577 Numbers that have only the digit "1" as first, central and final digit. For numbers with 5 or more digits the rest of digits are "0".

1, 111, 10101, 1001001, 100010001, 10000100001, 1000001000001, 100000010000001, 10000000100000001, 1000000001000000001, 100000000010000000001, 10000000000100000000001, 1000000000001000000000001, 100000000000010000000000001, 10000000000000100000000000001

Offset: 1

Author

Omar E. Pol, Feb 24 2008

Keywords

• nonn
• base
• less
• easy

Comments

Also, equal to A135576(n), written in base 2.

Essentially the same as A066138. - R. J. Mathar Apr 29 2008

a(n) has 2n-1 digits.

Examples

			----------------------------
n ............ a(n)
----------------------------
1 ............. 1
2 ............ 111
3 ........... 10101
4 .......... 1001001
5 ......... 100010001
6 ........ 10000100001
7 ....... 1000001000001
8 ...... 100000010000001
9 ..... 10000000100000001
10 ... 1000000001000000001
		

Links

• G. C. Greubel, Table of n, a(n) for n = 1..500
• Index entries for linear recurrences with constant coefficients, signature (111,-1110,1000).

Crossrefs

Cf. A001576, A066138, A135576, A138118, A138119, A138120, A138147, A138826.

Programs

Mathematica

Join[{1}, LinearRecurrence[{111, -1110, 1000}, {111, 10101, 1001001}, 25]] (* G. C. Greubel, Oct 19 2016 *)
Join[{1},Table[FromDigits[Join[{1},PadRight[{},n,0],{1},PadRight[{},n,0],{1}]],{n,0,10}]] (* Harvey P. Dale, Aug 15 2022 *)

PARI

Vec(-x*(2000*x^3-1110*x^2+1)/((x-1)*(10*x-1)*(100*x-1))  + O(x^100)) \\ Colin Barker, Sep 16 2013

Formula

a(n) = A135576(n), written in base 2.

Also, a(1)=1, for n>1; a(n)=(concatenation of 1, n-2 digits 0, 1, n-2 digits 0 and 1).

From Colin Barker, Sep 16 2013: (Start)

a(n) = 1 + 10^(n-1) + 100^(n-1) for n>1.

a(n) = 111*a(n-1) - 1110*a(n-2) + 1000*a(n-3) for n>4.

G.f.: x*(2000*x^3 - 1110*x^2 + 1)/((1-x)*(10*x-1)*(100*x-1)). (End)

E.g.f.: (-111 - 200*x + 100*exp(x) + 10*exp(10*x) + exp(100*x))/100. - Elmo R. Oliveira, Jun 13 2025