A138119 -id:A138119 - cp's OEIS frontend

A138146 Palindromes with 2n-1 digits formed from the reflected decimal expansion of the concatenation of 1, 1, 1 and infinite 0's.

1, 111, 11111, 1110111, 111000111, 11100000111, 1110000000111, 111000000000111, 11100000000000111, 1110000000000000111, 111000000000000000111, 11100000000000000000111

Offset: 1

Omar E. Pol, Mar 29 2008, May 18 2008

Bisection of A138145.

a(n) is also A147597(n) written in base 2. [Omar E. Pol, Nov 08 2008]

			n ............ a(n)
1 ............. 1
2 ............ 111
3 ........... 11111
4 .......... 1110111
5 ......... 111000111
6 ........ 11100000111
7 ....... 1110000000111
8 ...... 111000000000111
9 ..... 11100000000000111
10 ... 1110000000000000111

Index entries for linear recurrences with constant coefficients, signature (101,-100).

Cf. A138144, A138145.

Cf. A138118, A138119, A138120, A138721, A138826, A147597. [Omar E. Pol, Nov 08 2008]

Table[FromDigits@ If[n < 4, ConstantArray[1, 2 n - 1], Join[#, ConstantArray[0, 2 n - 7], #]] &@ ConstantArray[1, 3], {n, 14}] (* or *)
Rest@ CoefficientList[Series[-x (10 x - 1) (10 x + 1) (100 x^2 + 10 x + 1)/((x - 1) (100 x - 1)), {x, 0, 14}], x] (* Michael De Vlieger, Nov 25 2016 *)

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

From Colin Barker, Sep 16 2013: (Start)
a(n) = 111 + 111*100^(n-2) for n>3.
a(n) = 101*a(n-1) - 100*a(n-2) for n>5.
G.f.: -x*(10*x-1)*(10*x+1)*(100*x^2+10*x+1) / ((x-1)*(100*x-1)). (End)

Better definition from Omar E. Pol, Nov 16 2008

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".

Original entry on oeis.org

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

Offset: 1

Omar E. Pol, Feb 24 2008

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.

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

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

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

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 *)

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

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

A138145 Palindromes formed from the reflected decimal expansion of the concatenation of 1, 1, 1 and infinite 0's.

Original entry on oeis.org

1, 11, 111, 1111, 11111, 111111, 1110111, 11100111, 111000111, 1110000111, 11100000111, 111000000111, 1110000000111, 11100000000111, 111000000000111, 1110000000000111, 11100000000000111, 111000000000000111

Offset: 1

Omar E. Pol, Mar 29 2008

a(n) is also A147596(n) written in base 2. - Omar E. Pol, Nov 08 2008

			n .... a(n)
1 .... 1
2 .... 11
3 .... 111
4 .... 1111
5 .... 11111
6 .... 111111
7 .... 1110111
8 .... 11100111
9 .... 111000111
10 ... 1110000111
11 ... 11100000111
12 ... 111000000111
13 ... 1110000000111

Paolo Xausa, Table of n, a(n) for n = 1..995
Index entries for linear recurrences with constant coefficients, signature (11,-10).

Cf. A000533, A138144.

Cf. A138118, A138119, A138120, A138146, A138721, A138826, A147596. [From Omar E. Pol, Nov 08 2008]

Table[If[n < 7, (10^n - 1)/9, 111 + 111*10^(n-3)], {n, 25}] (* or *)
LinearRecurrence[{11, -10}, {1, 11, 111, 1111, 11111, 111111, 1110111}, 25] (* Paolo Xausa, Aug 08 2024 *)

Vec(-x*(10*x^2-1)*(100*x^4+10*x^2+1)/((x-1)*(10*x-1)) + O(x^100)) \\ Colin Barker, Sep 15 2013

From Colin Barker, Sep 15 2013: (Start)
a(n) = 111+111*10^(n-3) for n>5.
a(n) = 11*a(n-1)-10*a(n-2).
G.f.: -x*(10*x^2-1)*(100*x^4+10*x^2+1) / ((x-1)*(10*x-1)). (End)

Better definition from Omar E. Pol, Nov 16 2008

A138144 Palindromes formed from the reflected decimal expansion of the concatenation of 1, 1 and infinite 0's.

Original entry on oeis.org

1, 11, 111, 1111, 11011, 110011, 1100011, 11000011, 110000011, 1100000011, 11000000011, 110000000011, 1100000000011, 11000000000011, 110000000000011, 1100000000000011, 11000000000000011, 110000000000000011

Offset: 1

Omar E. Pol, Mar 29 2008

a(n) is also A147595(n) written in base 2. [From Omar E. Pol, Nov 08 2008]

			n .... a(n)
1 .... 1
2 .... 11
3 .... 111
4 .... 1111
5 .... 11011
6 .... 110011
7 .... 1100011
8 .... 11000011
9 .... 110000011
10 ... 1100000011

Index entries for linear recurrences with constant coefficients, signature (11,-10).

Cf. A000533.

Cf. A138118, A138119, A138120, A138145, A138146, A138721, A138826, A147595. [From Omar E. Pol, Nov 08 2008]

LinearRecurrence[{11,-10},{1,11,111,1111,11011},20] (* Harvey P. Dale, Aug 21 2016 *)

Vec(-x*(10*x^2-1)*(10*x^2+1)/((x-1)*(10*x-1)) + O(x^100)) \\ Colin Barker, Sep 15 2013

a(n) = 11+11*10^(n-2) for n>3. a(n) = 11*a(n-1)-10*a(n-2). G.f.: -x*(10*x^2-1)*(10*x^2+1) / ((x-1)*(10*x-1)). - Colin Barker, Sep 15 2013

Better definition. - Omar E. Pol, Nov 15 2008

A138147 Concatenation of n digits 1 and n digits 0.

Original entry on oeis.org

10, 1100, 111000, 11110000, 1111100000, 111111000000, 11111110000000, 1111111100000000, 111111111000000000, 11111111110000000000, 1111111111100000000000, 111111111111000000000000, 11111111111110000000000000, 1111111111111100000000000000, 111111111111111000000000000000

Offset: 1

Omar E. Pol, Mar 29 2008

Also, a(n) = binary representation of A020522(n), for n>0 (see example).

			n ... A020522(n) ..... a(n)
1 ....... 2 ........... 10
2 ...... 12 .......... 1100
3 ...... 56 ......... 111000
4 ..... 240 ........ 11110000
5 ..... 992 ....... 1111100000
6 .... 4032 ...... 111111000000
7 ... 16256 ..... 11111110000000

J.-P. Allouche and J. Shallit, Automatic Sequences, Cambridge Univ. Press, 2003, p. 136, Ex. 4.2.2. - N. J. A. Sloane, Jul 27 2012

Vincenzo Librandi, Table of n, a(n) for n = 1..95
Index entries for linear recurrences with constant coefficients, signature (110,-1000).

Cf. A002275, A020522, A138144, A138145, A138720, A276352.

Cf. A109241, A138118, A138119, A138120, A138146, A138721, A138826. - Omar E. Pol, Nov 08 2008

[(10^(2*n) - 10^n)/9: n in [1..30]]; // Vincenzo Librandi, Apr 26 2011

Table[FromDigits[Join[PadRight[{},n,1],PadRight[{},n,0]]],{n,15}] (* Harvey P. Dale, Nov 20 2011 *)

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

a(n) = (10^(2*n) - 10^n)/9 = A002275(n)*10^n. - Omar E. Pol, Apr 16 2008
a(n) = 10*A109241(n-1). - Omar E. Pol, Nov 08 2008
From Colin Barker, Sep 16 2013: (Start)
a(n) = 110*a(n-1) - 1000*a(n-2).
G.f.: 10*x/((10*x-1)*(100*x-1)). (End)
From Elmo R. Oliveira, Jun 13 2025: (Start)
E.g.f.: exp(10*x)*(exp(90*x) - 1)/9.
a(n) = A276352(n)/9. (End)

A147538 Numbers whose binary representation is the concatenation of n 1's and 2n-1 digits 0.

Original entry on oeis.org

2, 24, 224, 1920, 15872, 129024, 1040384, 8355840, 66977792, 536346624, 4292870144, 34351349760, 274844352512, 2198889037824, 17591649173504, 140735340871680, 1125891316908032, 9007164895002624, 72057456598974464, 576460202547609600

Offset: 1

Omar E. Pol, Nov 06 2008

a(n) is the number whose binary representation is A138119(n).

G. C. Greubel, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, signature (12,-32).

Cf. A138119.

Cf. A016152. - Omar E. Pol, Nov 13 2008

List([1..20], n-> 2^(2*n-1)*(2^n -1)); # G. C. Greubel, Jan 12 2020

[2^(2*n-1)*(2^n -1): n in [1..20]]; // G. C. Greubel, Jan 12 2020

seq(2^(2*n-1)*(2^n -1), n=1..20); # G. C. Greubel, Jan 12 2020

Table[FromDigits[Join[Table[1, {n}], Table[0, {2n - 1}]], 2], {n, 1, 20}] (* Stefan Steinerberger, Nov 11 2008 *)

vector(20, n, 2^(2*n-1)*(2^n -1)) \\ G. C. Greubel, Jan 12 2020

def a(n): return ((1 << n) - 1) << (2*n-1)
print([a(n) for n in range(1, 22)]) # Michael S. Branicky, Feb 24 2021

[2^(2*n-1)*(2^n -1) for n in (1..20)] # G. C. Greubel, Jan 12 2020

a(n) = 2^(2*n-1)*(2^n -1) = A081294(n)*A000225(n). - R. J. Mathar, Nov 09 2008
a(n) = 2*A016152(n). - Omar E. Pol, Nov 13 2008
From Colin Barker, Nov 04 2012: (Start)
a(n) = 12*a(n-1) - 32*a(n-2).
G.f.: 2*x/((1-4*x)*(1-8*x)). (End)

Extended by R. J. Mathar and Stefan Steinerberger, Nov 09 2008

A147816 Concatenation of n digits 1 and 2(n-1) digits 0.

Original entry on oeis.org

1, 1100, 1110000, 1111000000, 1111100000000, 1111110000000000, 1111111000000000000, 1111111100000000000000, 1111111110000000000000000, 1111111111000000000000000000, 1111111111100000000000000000000, 1111111111110000000000000000000000

Offset: 1

Omar E. Pol, Nov 13 2008

a(n) is also A016152(n) written in base 2.

			n ...... a(n)
1 ....... 1
2 ...... 1100
3 ..... 1110000
4 .... 1111000000
5 ... 1111100000000

Paolo Xausa, Table of n, a(n) for n = 1..300
Index entries for linear recurrences with constant coefficients, signature (1100,-100000).

Cf. A000533, A016152, A135577, A138119, A138120, A138144, A138145, A138146, A138721, A138826, A147757, A147759.

Array[(10^#-1)*10^(2*#-2)/9 &, 20] (* or *)
LinearRecurrence[{1100, -100000}, {1, 1100}, 20] (* Paolo Xausa, Feb 27 2024 *)

Vec(x/((100*x-1)*(1000*x-1))  + O(x^100)) \\ Colin Barker, Sep 16 2013

a(n) = A138119(n)/10.

a(n) = 1100*a(n-1)-100000*a(n-2). G.f.: x / ((100*x-1)*(1000*x-1)). - Colin Barker, Sep 16 2013

A369405 Context-free language 1^n.0^(2n).

Original entry on oeis.org

100, 110000, 111000000, 111100000000, 111110000000000, 111111000000000000, 111111100000000000000, 111111110000000000000000, 111111111000000000000000000, 111111111100000000000000000000, 111111111110000000000000000000000, 111111111111000000000000000000000000

Offset: 1

Vyom Narsana, Jan 22 2024

This sequence represents the context-free language 1^n.0^(2n) which can be accepted by a pushdown automaton. It finds applications in the study of formal languages and automata theory in theoretical computer science.

Paolo Xausa, Table of n, a(n) for n = 1..300
Index entries for linear recurrences with constant coefficients, signature (1100,-100000).

Cf. A007088, A059409, A138119, A147816.

a:= n-> convert(4^n*(2^n-1), binary):
seq(a(n), n=1..15);  # Alois P. Heinz, Feb 04 2024

Array[(10^#-1)*10^(2*#)/9 &, 20] (* or *)
LinearRecurrence[{1100, -100000}, {100, 110000}, 20] (* Paolo Xausa, Feb 27 2024 *)

def A369405(n): return (10**n-1)//9*10**(n<<1) # Chai Wah Wu, Feb 11 2024

From Robert Israel, Jan 22 2024: (Start)
a(n) = (10^n-1)*10^(2*n)/9.
G.f.: 100*x/(100000*x^2 - 1100*x + 1). (End)
From Alois P. Heinz, Feb 04 2024: (Start)
a(n) = A007088(A059409(n)).
a(n) = 10 * A138119(n).
a(n) = 100 * A147816(n). (End)

cp's OEIS Frontend

A138146 Palindromes with 2n-1 digits formed from the reflected decimal expansion of the concatenation of 1, 1, 1 and infinite 0's.

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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".

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A138145 Palindromes formed from the reflected decimal expansion of the concatenation of 1, 1, 1 and infinite 0's.

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A138144 Palindromes formed from the reflected decimal expansion of the concatenation of 1, 1 and infinite 0's.

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A138147 Concatenation of n digits 1 and n digits 0.

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A147538 Numbers whose binary representation is the concatenation of n 1's and 2n-1 digits 0.

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