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

Offset

1,2

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.

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).

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)

Example

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

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 *)

Prog

(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

Crossrefs

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

Sequence in context: A118109 A265280 A267275 * A153498 A118110 A267608

Adjacent sequences: A135574 A135575 A135576 * A135578 A135579 A135580

Keyword

nonn,base,less,easy

Author

Omar E. Pol, Feb 24 2008

Status

approved