A138118 -id:A138118 - 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

A147590 Numbers whose binary representation is the concatenation of 2n-1 digits 1 and n-1 digits 0.

Original entry on oeis.org

1, 14, 124, 1016, 8176, 65504, 524224, 4194176, 33554176, 268434944, 2147482624, 17179867136, 137438949376, 1099511619584, 8796093005824, 70368744144896, 562949953355776, 4503599627239424, 36028797018701824, 288230376151187456

Offset: 1

Omar E. Pol, Nov 08 2008

a(n) is the number whose binary representation is A147589(n).

			     1_10 is 1_2;
    14_10 is 1110_2;
   124_10 is 1111100_2;
  1016_10 is 1111111000_2.

Nathaniel Johnston, Table of n, a(n) for n = 1..500
Index entries for linear recurrences with constant coefficients, signature (10,-16).

Cf. A020540, A138118, A147537, A147589.

List([1..25], n-> 2^(n-2)*(4^n-2)); # G. C. Greubel, Jul 27 2019

[8^n/4-2^(n-1): n in [1..25]]; // Vincenzo Librandi, Jul 27 2019

seq(8^n/4-2^(n-1),n=1..25); # Nathaniel Johnston, Apr 30 2011

LinearRecurrence[{10,-16},{1,14},30] (* Harvey P. Dale, Oct 10 2014 *)
Table[8^n / 4 - 2^(n - 1), {n, 25}] (* Vincenzo Librandi, Jul 27 2019 *)

vector(25, n, 2^(n-2)*(4^n-2)) \\ G. C. Greubel, Jul 27 2019

[2^(n-2)*(4^n-2) for n in (1..25)] # G. C. Greubel, Jul 27 2019

a(n) = A147537(n)/2.
From R. J. Mathar, Jul 13 2009: (Start)
a(n) = 8^n/4 - 2^(n-1) = A083332(2n-2).
a(n) = 10*a(n-1) - 16*a(n-2).
G.f.: x*(1+4*x)/((1-2*x)*(1-8*x)). (End)
From César Aguilera, Jul 26 2019: (Start)
Lim_{n->infinity} a(n)/a(n-1) = 8;
a(n)/a(n-1) = 8 + 6/A083420(n). (End)
E.g.f.: (1/4)*(exp(2*x)*(-2 + exp(6*x)) + 1). - Stefano Spezia, Aug 05 2019
a(n) = A020540(n - 1)/4. - Jon Maiga, Aug 05 2019

More terms from R. J. Mathar, Jul 13 2009

Typo in a(12) corrected by Omar E. Pol, Jul 20 2009

A138119 Concatenation of n digits 1 and 2*n-1 digits 0.

Original entry on oeis.org

10, 11000, 11100000, 11110000000, 11111000000000, 11111100000000000, 11111110000000000000, 11111111000000000000000, 11111111100000000000000000, 11111111110000000000000000000, 11111111111000000000000000000000, 11111111111100000000000000000000000

Offset: 1

Omar E. Pol, Apr 03 2008

a(n) has 3*n-1 digits.

a(n) is also A147538(n) written in base 2. - Omar E. Pol, Nov 08 2008.

			n ...... a(n)
1 ....... 10
2 ...... 11000
3 ..... 11100000
4 .... 11110000000
5 ... 11111000000000

Index entries for linear recurrences with constant coefficients, signature (1100,-100000).

Cf. A138118, A138145, A138146, A138147, A138719, A138720, A138721, A138722, A147538.

LinearRecurrence[{1100, -100000}, {10, 11000}, 15] (* Paolo Xausa, Feb 06 2024 *)

Vec(10*x/((100*x-1)*(1000*x-1)) + O(x^100)) \\ Colin Barker, Sep 16 2013

From Colin Barker, Sep 16 2013: (Start)
a(n) = 1100*a(n-1) - 100000*a(n-2).
G.f.: 10*x / ((100*x-1)*(1000*x-1)). (End)

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)

A147537 Numbers whose binary representation is the concatenation of 2n-1 digits 1 and n digits 0.

Original entry on oeis.org

2, 28, 248, 2032, 16352, 131008, 1048448, 8388352, 67108352, 536869888, 4294965248, 34359734272, 274877898752, 2199023239168, 17592186011648, 140737488289792, 1125899906711552, 9007199254478848, 72057594037403648, 576460752302374912, 4611686018425290752

Offset: 1

Omar E. Pol, Nov 06 2008

a(n) is the number whose binary representation is A138118(n).

G. C. Greubel, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, signature (10,-16).

Cf. A138118.

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

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

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

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

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

def a(n): return ((1 << (2*n-1)) - 1) << n
print([a(n) for n in range(1, 22)]) # Michael S. Branicky, Feb 28 2021

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

From Colin Barker, Nov 04 2012: (Start)
a(n) = 2^(n-1)*(4^n - 2) = 2*A147590(n).
a(n) = 10*a(n-1) - 16*a(n-2).
G.f.: 2*x*(1+4*x)/((1-2*x)*(1-8*x)). (End)

A138306 Least prime p such that 2n = p + g, where g is a prime primitive root of p, or 0 if there is no such prime p.

Original entry on oeis.org

0, 0, 0, 5, 7, 7, 0, 0, 11, 13, 17, 13, 0, 17, 23, 19, 23, 0, 0, 23, 23, 31, 43, 29, 37, 41, 37, 37, 41, 41, 43, 0, 47, 61, 41, 43, 67, 47, 47, 67, 59, 53, 73, 47, 47, 61, 53, 59, 67, 0, 59, 61, 59, 67, 97, 89, 97, 79, 71, 61, 79, 71, 73, 67, 71, 71, 97, 83, 71, 137, 113, 0, 103, 89

Offset: 1

T. D. Noe, Mar 14 2008

nonn

Goldbach meets prime primitive prime roots. Sequence A138118 gives the number of representations for each 2n and sequence A138307 lists the values of 2n that are not represented.

T. D. Noe, Table of n, a(n) for n=1..1000

Cf. A138305 (prime primitive roots).

A147589 Concatenation of 2n-1 digits 1 and n-1 digits 0.

Original entry on oeis.org

1, 1110, 1111100, 1111111000, 1111111110000, 1111111111100000, 1111111111111000000, 1111111111111110000000, 1111111111111111100000000, 1111111111111111111000000000, 1111111111111111111110000000000

Offset: 1

Omar E. Pol, Nov 08 2008

a(n) is also A147590(n) written in base 2.

			n .......... a(n)
1 ........... 1
2 ......... 1110
3 ....... 1111100
4 ..... 1111111000
5 ... 1111111110000

Index entries for linear recurrences with constant coefficients, signature (1010,-10000).

Cf. A138118, A147537, A147590.

vector(100, n, 10^(-2+n)*(-10+100^n)/9) \\ Colin Barker, Jul 08 2014

Vec(x*(100*x+1)/((10*x-1)*(1000*x-1)) + O(x^100)) \\ Colin Barker, Jul 08 2014

a(n) = A138118(n)/10.
a(n) = {[10^(2*n-1)-1]*10^(n-1)}/9, with n>=1. - Paolo P. Lava, Nov 26 2008
G.f.: x*(100*x+1) / ((10*x-1)*(1000*x-1)). - Colin Barker, Jul 08 2014

Keyword:base added by Charles R Greathouse IV, Apr 28 2010

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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A147590 Numbers whose binary representation is the concatenation of 2n-1 digits 1 and n-1 digits 0.

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A138119 Concatenation of n digits 1 and 2*n-1 digits 0.

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