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 1-7 of 7 results.

A153498 Palindromes formed from concatenation of A147759(n) and the same string A147759(n) but without its initial digit.

Original entry on oeis.org

1, 111, 10101, 1001001, 101010101, 10110101101, 1010101010101, 101001010100101, 10101010101010101, 1010110101010110101, 101010101010101010101, 10101001010101010010101
Offset: 1

Views

Author

Omar E. Pol, Dec 27 2008, Feb 18 2009

Keywords

Comments

a(n) is also A153497(n) written in base 2.

Examples

			n ............. a(n)
1 .............. 1
2 ............. 111
3 ............ 10101
4 ........... 1001001
5 .......... 101010101
6 ......... 10110101101
7 ........ 1010101010101
8 ....... 101001010100101
9 ...... 10101010101010101
10 .... 1010110101010110101
11 ... 101010101010101010101
======================================
Another visualization of the structure
======================================
1 .............. *
2 ............. /|\
3 ............ /.|.\
4 ........... /..|..\
5 .......... /.*.|.*.\
6 ......... /./|.|.|\.\
7 ........ /./.|.|.|.\.\
8 ....... /./..|.|.|..\.\
9 ...... /./.*.|.|.|.*.\.\
10 .... /././|.|.|.|.|\.\.\
11 ... /././.|.|.|.|.|.\.\.\

Links

Crossrefs

Cf. A135577, A138146, A152576, A152764, A153497, A153499, A153500, A144564.

Formula

From R. J. Mathar, Feb 20 2009: (Start)
a(n)=101*a(n-1)-1110*a(n-2)+102010*a(n-3)-111000*a(n-4)+1010000*a(n-5)-1000000*a(n-6).
G.f.: x(1+10x+2000x^3-91000*x^4+100000x^5)/((1-100x)(1-x)(1+10x^2)(1+1000x^2)). (End)

Extensions

More terms from R. J. Mathar, Feb 20 2009
Keyword:base added by Charles R Greathouse IV, Apr 23 2010

A153500 First 3 terms coincide with A152756. For n>3, a(n) is the palindromic number formed from concatenation of 1, 0, A147759(n-3), 0, A147759(n-3), 0 and 1.

Original entry on oeis.org

1, 101, 10001, 1010101, 101101101, 10101010101, 1010010100101, 101010101010101, 10101101010110101, 1010101010101010101, 101010010101010010101, 10101010101010101010101, 1010101101010101011010101, 101010101010101010101010101, 10101010010101010101001010101
Offset: 1

Views

Author

Omar E. Pol, Dec 27 2008, Feb 18 2009

Keywords

Comments

a(n) is also A153499(n) written in base 2.

Examples

			n ............ a(n)
1 ............. 1
2 ............ 101
3 ........... 10001
4 .......... 1010101
5 ......... 101101101
6 ........ 10101010101
7 ....... 1010010100101
8 ...... 101010101010101
9 ..... 10101101010110101
10 ... 1010101010101010101
======================================
Another visualization of the structure
======================================
1 ............. *
2 ............ /.\
3 ........... /...\
4 .......... /.*.*.\
5 ......... /./|.|\.\
6 ........ /./.|.|.\.\
7 ....... /./..|.|..\.\
8 ...... /./.*.|.|.*.\.\
9 ..... /././|.|.|.|\.\.\
10 ... /././.|.|.|.|.\.\.\

Links

Crossrefs

Cf. A135577, A138146, A152576, A152764, A153497, A153498, A153499, A144564.

Formula

a(n) = 101*a(n-1)-1110*a(n-2)+102010*a(n-3)-111000*a(n-4)+1010000*a(n-5)-1000000*a(n-6), n>7. [R. J. Mathar, Feb 20 2009]
G.f.: -x*(1000000*x^6-1010000*x^5+10000*x^4-10100*x^3-910*x^2-1) / ((x-1)*(100*x-1)*(10*x^2+1)*(1000*x^2+1)). [Colin Barker, Sep 17 2013]

Extensions

More terms from R. J. Mathar, Feb 20 2009
Keyword:base added by Charles R Greathouse IV, Apr 26 2010
More terms from Colin Barker, Sep 17 2013

A094028 Expansion of 1/((1-x)*(1-100*x)).

Original entry on oeis.org

1, 101, 10101, 1010101, 101010101, 10101010101, 1010101010101, 101010101010101, 10101010101010101, 1010101010101010101, 101010101010101010101, 10101010101010101010101, 1010101010101010101010101, 101010101010101010101010101, 10101010101010101010101010101
Offset: 0

Views

Author

Paul Barry, Apr 22 2004

Keywords

Comments

Regarded as binary numbers and converted to decimal, these become 1,5,21,85,... the partial sums of 4^n (see A002450).
Partial sums of 100^n.
Odd terms of A056830. - Alexandre Wajnberg, May 31 2005
101 is the only term that is prime, since (100^k-1)/99 = (10^k+1)/11 * (10^k-1)/9. When k is odd and not 1, (10^k+1)/11 is an integer > 1 and thus (100^k-1)/99 is nonprime. When k is even and greater than 2, (100^k-1)/99 has the prime factor 101 and is nonprime. - Felix Fröhlich, Oct 17 2015
Previous comment is the answer to the problem A1 proposed during the 50th Putnam Competition in 1989 (link). - Bernard Schott, Mar 24 2023

Examples

			From _Omar E. Pol_, Dec 13 2008: (Start)
=======================
n ....... a(n)
0 ........ 1
1 ....... 101
2 ...... 10101
3 ..... 1010101
4 .... 101010101
5 ... 10101010101
======================
(End)

References

  • Clifford A. Pickover, A Passion for Mathematics, Wiley, 2005; see p. 60.
  • Stephen Wolfram, A New Kind of Science, Wolfram Media, 2002; p. 55.

Links

Crossrefs

Bisection of A147759. [Omar E. Pol, Nov 13 2008]
Cf. A002450, A056830, A094027.
Cf. similar sequences of the form (k^n-1)/(k-1) listed in A269025.

Programs

  • Magma
    [1+100*(100^n-1)/99 : n in [0..15]]; // Wesley Ivan Hurt, Oct 17 2015
  • Maple
    A094028:=n->1+100*(100^n-1)/99: seq(A094028(n), n=0..15); # Wesley Ivan Hurt, Oct 17 2015
  • Mathematica
    CoefficientList[Series[1/((1-x)(1-100x)),{x,0,20}],x] (* or *) Table[ FromDigits[ PadRight[{},2n-1,{1,0}]],{n,20}] (* or *) LinearRecurrence[ {101,-100},{1,101},20] (* or *) NestList[100#+1&,1,20] (* Harvey P. Dale, Apr 27 2015 *)
  • Maxima
    A094028(n):=1+100*(100^n-1)/99$
makelist(A094028(n),n,0,30); /* Martin Ettl, Nov 06 2012 */
  • PARI
    a(n) = 1+100*(100^n-1)/99 \\ Felix Fröhlich, Oct 17 2015
  • PARI
    Vec(1/((1-x)*(1-100*x)) + O(x^100)) \\ Altug Alkan, Oct 17 2015

Formula

G.f.: 1/((1-x)*(1-100*x)).
a(n) = 1 + 100*(100^n-1)/99. - N. J. A. Sloane, Apr 20 2008
a(n) = 100^(n+1)/99 - 1/99.
a(n) = A094027(2*n+1).
a(n) = 100*a(n-1) + 1, a(0) = 1. - Philippe Deléham, Feb 22 2014
a(n) = 101*a(n-1) - 100*a(n-2) for n > 1. - Wesley Ivan Hurt, Oct 17 2015
a(n) = (100^(n+1) - 1)/99. - Bernard Schott, Apr 15 2021
E.g.f.: exp(x)*(100*exp(99*x) - 1)/99. - Elmo R. Oliveira, Mar 06 2025

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

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

A144863 Start with 1, then at each step prepend 10 and append 01.

Original entry on oeis.org

1, 10101, 101010101, 1010101010101, 10101010101010101, 101010101010101010101, 1010101010101010101010101, 10101010101010101010101010101, 101010101010101010101010101010101
Offset: 1

Views

Author

Artur Jasinski, Sep 23 2008, Sep 25 2008

Keywords

Comments

Bisection of A094028. - Omar E. Pol, Nov 12 2008
a(n) is also A144864(n) written in base 2. - Omar E. Pol, Nov 13 2008
Quadrisection of A147759. - Omar E. Pol, Nov 16 2008

Links

Crossrefs

Cf. A056830, A094028, A135576, A144864, A147759.

Programs

  • Mathematica
    a = {}; k = {1}; Do[x = FromDigits[k, 2]; AppendTo[a, FromDigits[RealDigits[x, 2]]]; AppendTo[k, 0]; AppendTo[k, 1]; PrependTo[k, 0]; PrependTo[k, 1], {n, 1, 100}];
Table[FromDigits[RealDigits[1/12 (-4 + 16^n), 2]], {n, 1, 10}]
a = {}; k = 1; Do[AppendTo[a, k]; k = 10000 k + 101, {n, 1, 10}]; a
Table[1/99 (-1 + 100^(-1 + 2 n)), {n, 1, 20}]
LinearRecurrence[{10001,-10000},{1,10101},20] (* Harvey P. Dale, Aug 22 2014 *)

Formula

a(n) = (-1+100^(-1+2*n))/99.
If a(n) is interpreted as binary number, (-4+16^n)/12 gives the decimal representation of a(n).
a(n) = 10000*a(n-1)+101, n>1.
G.f.: x*(1+100*x) / ( (10000*x-1)*(x-1) ).

A239577 Expansion of 1/((x-1)*(3*x-1)*(3*x^2+1)).

Original entry on oeis.org

1, 4, 10, 28, 91, 280, 820, 2440, 7381, 22204, 66430, 199108, 597871, 1794160, 5380840, 16140880, 48427561, 145287604, 435848050, 1307529388, 3922632451, 11767941640, 35303692060, 105910943320, 317733228541, 953200084204, 2859599056870, 8578795974868
Offset: 0

Views

Author

Philippe Deléham, Mar 21 2014

Keywords

Examples

			Ternary................Decimal
1............................1
11...........................4
101.........................10
1001........................28
10101.......................91
101101.....................280
1010101....................820
10100101..................2440
101010101.................7381
1010110101...............22204
10101010101..............66430
101010010101............199108, etc.

Links

Crossrefs

Cf. A002452, A015251, A147759, A154957.

Programs

  • Mathematica
    Table[(-1 + 3^(2 + n) + (-1 + (-1)^n) (-3)^((1 + n)/2))/8, {n, 0, 30}] (* Bruno Berselli, Mar 24 2014 *)
CoefficientList[Series[1/((x - 1) (3 x - 1) (3 x^2 + 1)), {x, 0, 40}], x] (* Vincenzo Librandi, Mar 24 2014 *)
LinearRecurrence[{4,-6,12,-9},{1,4,10,28},30] (* Harvey P. Dale, Oct 04 2024 *)

Formula

G.f.: 1/((x-1)*(3*x-1)*(3*x^2+1)).
a(n) = Sum{k=0..n} A154957(n,k)*3^k.
a(n) = 4*a(n-1) - 6*a(n-2) + 12*a(n-3) - 9*a(n-4) for n > 3, a(0)=1, a(1)=4, a(2)=10, a(3)=16.
a(2*n) = A002452(n+1); a(2*n+1) = 4*A015251(n+2).
a(n) = ( -1 + 3^(2+n) + (-1+(-1)^n)*(-3)^((1+n)/2) )/8. [Bruno Berselli, Mar 24 2014]
Showing 1-7 of 7 results.

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