cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-6 of 6 results.

A144564 Bisection of A147757.

Original entry on oeis.org

1, 101, 10101, 1011101, 101111101, 10111111101, 1011111111101, 101111111111101, 10111111111111101, 1011111111111111101, 101111111111111111101, 10111111111111111111101, 1011111111111111111111101, 101111111111111111111111101, 10111111111111111111111111101
Offset: 1

Views

Author

Omar E. Pol, Dec 14 2008

Keywords

Examples

			n ...... a(n)
1 ....... 1
2 ...... 101
3 ..... 10101
4 .... 1011101
5 ... 101111101

Links

Crossrefs

Cf. A135577, A138120, A138146, A147757, A152756.

Programs

  • Mathematica
    Rest[CoefficientList[Series[x(1+10x)(100x^2-10x+1)/((100x-1)(x-1)),{x,0,20}],x]] (* or *) Join[{1,101},Table[FromDigits[Join[{1,0},PadRight[ {},2n+1,1],{0,1}]],{n,0,20}]] (* Harvey P. Dale, Dec 26 2014 *)

Formula

G.f.: x*(1+10*x)*(100*x^2-10*x+1)/((100*x-1)*(x-1)). - R. J. Mathar, Aug 24 2011

A138721 Concatenation of n digits 1, n digits 0 and n digits 1.

Original entry on oeis.org

101, 110011, 111000111, 111100001111, 111110000011111, 111111000000111111, 111111100000001111111, 111111110000000011111111, 111111111000000000111111111, 111111111100000000001111111111, 111111111110000000000011111111111, 111111111111000000000000111111111111
Offset: 1

Views

Author

Omar E. Pol, Mar 29 2008

Keywords

Comments

a(n) is also A145641(n) written in base 2. - Omar E. Pol, Oct 15 2008
a(n) has 3n digits. - Omar E. Pol, Nov 12 2008

Examples

			From _Omar E. Pol_, Nov 12 2008: (Start)
n         Successive digits of a(n)
1                 ( 1 0 1 )
2              ( 1 1 0 0 1 1 )
3           ( 1 1 1 0 0 0 1 1 1 )
4        ( 1 1 1 1 0 0 0 0 1 1 1 1 )
5     ( 1 1 1 1 1 0 0 0 0 0 1 1 1 1 1 )
(End)

Links

Crossrefs

Cf. A000533, A135577, A138120, A138144, A138145, A138146, A138826, A145641, A147757, A147759.

Programs

  • Maple
    a:= n-> parse(cat(1$n,0$n,1$n)):
seq(a(n), n=1..14);  # Alois P. Heinz, Mar 03 2022
  • Mathematica
    Table[(100^n + 1)*(10^n - 1)/9, {n, 15}] (* Paolo Xausa, Aug 02 2024 *)
  • PARI
    Vec(x*(101000*x^2-2200*x+101)/((x-1)*(10*x-1)*(100*x-1)*(1000*x-1)) + O(x^100)) \\ Colin Barker, Sep 16 2013

Formula

G.f.: x*(101000*x^2 - 2200*x + 101) / ((x-1)*(10*x-1)*(100*x-1)*(1000*x-1)). - Colin Barker, Sep 16 2013
a(n) = (100^n+1)*(10^n-1)/9. - Paolo Xausa, Aug 02 2024

A147759 Palindromes formed from the reflected decimal expansion of the infinite concatenation of 1's and 0's.

Original entry on oeis.org

1, 11, 101, 1001, 10101, 101101, 1010101, 10100101, 101010101, 1010110101, 10101010101, 101010010101, 1010101010101, 10101011010101, 101010101010101, 1010101001010101, 10101010101010101, 101010101101010101
Offset: 1

Views

Author

Omar E. Pol, Nov 11 2008

Keywords

Comments

a(k(n)) is divisible by 3 iff k(n) is defined by k(1) = 5 and k(n+1) - k(n) = A100285(n+2). - Altug Alkan, Dec 05 2015

Examples

			n .... Successive digits of a(n)
1 ............. ( 1 )
2 ............ ( 1 1 )
3 ........... ( 1 0 1 )
4 .......... ( 1 0 0 1 )
5 ......... ( 1 0 1 0 1 )
6 ........ ( 1 0 1 1 0 1 )
7 ....... ( 1 0 1 0 1 0 1 )
8 ...... ( 1 0 1 0 0 1 0 1 )
9 ..... ( 1 0 1 0 1 0 1 0 1 )
10 ... ( 1 0 1 0 1 1 0 1 0 1 )

Links

Crossrefs

Cf. A000533, A094028, A135577, A138120, A138144, A138145, A138146, A138721, A138826, A147757.

Programs

  • Magma
    I:=[1,11,101,1001]; [n le 4 select I[n] else 11*Self(n-1)-20*Self(n-2)+110*Self(n-3)-100*Self(n-4): n in [1..30]]; // Vincenzo Librandi, Dec 05 2015
  • Mathematica
    CoefficientList[Series[x/((1 - x) (1 - 10 x) (1 + 10 x^2)),{x, 0, 20}], x] (* Vincenzo Librandi, Dec 05 2015 *)
LinearRecurrence[{11,-20,110,-100},{1,11,101,1001},30] (* Harvey P. Dale, Apr 10 2022 *)
  • PARI
    Vec(x/((1-x)*(1-10*x)*(1+10*x^2)) + O(x^30)) \\ Michel Marcus, Dec 05 2015

Formula

From R. J. Mathar, Feb 20 2009: (Start)
a(n) = 11*a(n-1)-20*a(n-2)+110*a(n-3)-100*a(n-4).
G.f.: x/((1-x)*(1-10*x)*(1+10*x^2)). (End)
E.g.f.: (exp(x)*(10*exp(9*x) - 1) - 9*cos(sqrt(10)*x))/99. - Stefano Spezia, Oct 12 2024

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

Views

Author

Omar E. Pol, Nov 13 2008

Keywords

Comments

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

Examples

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

Links

Crossrefs

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

Programs

  • Mathematica
    Array[(10^#-1)*10^(2*#-2)/9 &, 20] (* or *)
LinearRecurrence[{1100, -100000}, {1, 1100}, 20] (* Paolo Xausa, Feb 27 2024 *)
  • PARI
    Vec(x/((100*x-1)*(1000*x-1))  + O(x^100)) \\ Colin Barker, Sep 16 2013

Formula

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

A147758 Numbers whose binary representation is a palindrome formed from the reflected decimal expansion of the concatenation of 1, 0 and infinite digits 1.

Original entry on oeis.org

1, 3, 5, 9, 21, 45, 93, 189, 381, 765, 1533, 3069, 6141, 12285, 24573, 49149, 98301, 196605, 393213, 786429, 1572861, 3145725, 6291453, 12582909, 25165821, 50331645, 100663293, 201326589, 402653181, 805306365, 1610612733, 3221225469
Offset: 1

Views

Author

Omar E. Pol, Nov 11 2008

Keywords

Comments

a(n) is the number whose binary representation is A147757(n).

Crossrefs

Cf. A068156, A147757.

Extensions

More terms from Sean A. Irvine, Nov 26 2009

A279389 3 times Mersenne primes A000668.

Original entry on oeis.org

9, 21, 93, 381, 24573, 393213, 1572861, 6442450941, 6917529027641081853, 1856910058928070412348686333, 486777830487640090174734030864381, 510423550381407695195061911147652317181
Offset: 1

Views

Author

Omar E. Pol, Dec 20 2016

Keywords

Comments

Also sum of n-th Mersenne prime and the radical of n-th even perfect number.
The binary representation of a(n) has only two zeros, starting with "10" and ending with "01". The sequence begins: 1001, 10101, 1011101, 101111101, 101111111111101,...

Crossrefs

Cf. A000010, A000203, A000396, A000668, A051027, A147757.
Subsequence of A001748, and of A147758, and of A174055, and possibly of other sequences, see below:
Cf. A060482, A068156, A072087, A136252, A144482, A144856, A249780, A269633.

Formula

a(n) = 3*A000668(n) = A000668(n) + A139257(n).
a(n) = phi(M(n)) + sigma(sigma(M(n))) = A000010(A000668(n)) + A000203(A000203(A000668(n))) = A000010(A000668(n)) + A051027(A000668(n)).
Showing 1-6 of 6 results.

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