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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". 13
1, 111, 10101, 1001001, 100010001, 10000100001, 1000001000001, 100000010000001, 10000000100000001, 1000000001000000001, 100000000010000000001, 10000000000100000000001 (list; graph; refs; listen; history; text; internal format)
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).

a(n) = 10^(2*n-2) + 10^(n-1) + 1 - 2*(C(2*(n-1),n-1) mod 2). - Paolo P. Lava, Nov 26 2008

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 *)

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

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Last modified November 20 10:41 EST 2017. Contains 294963 sequences.