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 1-3 of 3 results.

A147757 Palindromes formed from the reflected decimal expansion of the concatenation of 1, 0 and infinite digits 1.

Original entry on oeis.org

1, 11, 101, 1001, 10101, 101101, 1011101, 10111101, 101111101, 1011111101, 10111111101, 101111111101, 1011111111101, 10111111111101, 101111111111101, 1011111111111101, 10111111111111101, 101111111111111101
Offset: 1

Views

Author

Omar E. Pol, Nov 11 2008

Keywords

Comments

a(n) is also A147758(n) written in base 2.
a(A016789(n)) is divisible by 3 for n > 0. - Altug Alkan, Dec 06 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 1 1 0 1 )
8 ...... ( 1 0 1 1 1 1 0 1 )
9 ..... ( 1 0 1 1 1 1 1 0 1 )
10 ... ( 1 0 1 1 1 1 1 1 0 1 )

Links

Crossrefs

Cf. A000533, A016789, A135577, A138120, A138144, A138145, A138146, A138721, A138826, A147758.
Cf. A144564 (a bisection).

Programs

  • Mathematica
    f[n_] := Block[{w = {1, 0}}, Which[n == 1, w = {1}, n == 2, w = {1, 1}, n == 3, AppendTo[w, 1], n >= 4, w = Join[w, Table[1, {n - 4}], Reverse@ w]]; FromDigits@ w]; Array[f, 19] (* Michael De Vlieger, Dec 05 2015 *)
LinearRecurrence[{11,-10},{1,11,101,1001,10101},20] (* Harvey P. Dale, Aug 02 2017 *)
  • PARI
    Vec( x+11*x^2+101*x^3 -91*x^4*(-11+10*x) / ( (10*x-1)*(x-1) ) + O(x^30)) \\ Michel Marcus, Dec 05 2015

Formula

G.f.: x+11*x^2+101*x^3-91*x^4*(-11+10*x) / ( (10*x-1)*(x-1) ). - R. J. Mathar, Aug 24 2011
a(n) = 11*a(n-1) - 10*a(n-2) for n>2. Wesley Ivan Hurt, Dec 06 2015

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
Showing 1-3 of 3 results.

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