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.

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

Views

Author

Omar E. Pol, Mar 29 2008

Keywords

Comments

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

Examples

			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

Links

Crossrefs

Cf. A000533, A138144.
Cf. A138118, A138119, A138120, A138146, A138721, A138826, A147596. [From Omar E. Pol, Nov 08 2008]

Programs

  • Mathematica
    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 *)
  • PARI
    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

Formula

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)

Extensions

Better definition from Omar E. Pol, Nov 16 2008

A147595 a(n) is the number whose binary representation is A138144(n).

Original entry on oeis.org

1, 3, 7, 15, 27, 51, 99, 195, 387, 771, 1539, 3075, 6147, 12291, 24579, 49155, 98307, 196611, 393219, 786435, 1572867, 3145731, 6291459, 12582915, 25165827, 50331651, 100663299, 201326595, 402653187, 805306371, 1610612739, 3221225475
Offset: 1

Views

Author

Omar E. Pol, Nov 08 2008

Keywords

Links

Crossrefs

Cf. A138144, A145641, A147537, A147538, A147539, A147540, A147590, A147596, A147597.

Programs

  • Magma
    [1,3,7] cat [3*(1+2^(n-2)): n in [4..40]]; // G. C. Greubel, Oct 25 2022
  • Mathematica
    LinearRecurrence[{3,-2},{1,3,7,15,27},40] (* Harvey P. Dale, Nov 30 2020 *)
  • PARI
    Vec(-x*(2*x^2-1)*(2*x^2+1)/((x-1)*(2*x-1)) + O(x^100)) \\ Colin Barker, Sep 15 2013
  • SageMath
    [1,3,7]+[3*(1+2^(n-2)) for n in range(4,40)] # G. C. Greubel, Oct 25 2022

Formula

a(n) = A060013(n+2), n > 3. - R. J. Mathar, Feb 05 2010
a(n+4) = 3*(2^(n+2) + 1), n >= 0. - Brad Clardy, Apr 03 2013
From Colin Barker, Sep 15 2013: (Start)
a(n) = 3*(4 + 2^n)/4 for n>3.
a(n) = 3*a(n-1) - 2*a(n-2).
G.f.: x*(1-2*x^2)*(1+2*x^2) / ((1-x)*(1-2*x)). (End)
E.g.f.: (3/4)*(4*exp(x) + exp(2*x)) - (15/4) - 7*x/2 - 3*x^2/2 - x^3/3. - G. C. Greubel, Oct 25 2022

Extensions

Extended by R. J. Mathar, Feb 05 2010

A147597 a(n) is the number whose binary representation is A138146(n).

Original entry on oeis.org

1, 7, 31, 119, 455, 1799, 7175, 28679, 114695, 458759, 1835015, 7340039, 29360135, 117440519, 469762055, 1879048199, 7516192775, 30064771079, 120259084295, 481036337159, 1924145348615, 7696581394439, 30786325577735, 123145302310919, 492581209243655
Offset: 1

Views

Author

Omar E. Pol, Nov 08 2008

Keywords

Comments

Bisection of A147596.

Links

Crossrefs

Cf. A138146, A145641, A147537, A147538, A147539.
Cf. A147540, A147590, A147595, A147596.

Programs

  • Magma
    [1,7,31] cat [7*(1+4^(n-2)): n in [4..40]]; // G. C. Greubel, Oct 25 2022
  • Mathematica
    Table[FromDigits[#, 2] &@ If[n < 4, ConstantArray[1, 2 n - 1], Join[#, ConstantArray[0, 2 n - 7], #]] &@ ConstantArray[1, 3], {n, 25}] (* or *)
Rest@ CoefficientList[Series[x (2 x + 1) (2 x - 1) (4 x^2 + 2 x + 1)/((4 x - 1) (1 - x)), {x, 0, 25}], x] (* Michael De Vlieger, Nov 25 2016 *)
LinearRecurrence[{5,-4},{1,7,31,119,455},30] (* Harvey P. Dale, Aug 04 2025 *)
  • PARI
    Vec(x*(2*x+1)*(2*x-1)*(4*x^2+2*x+1)/((4*x-1)*(1-x)) + O(x^30)) \\ Colin Barker, Nov 25 2016
  • SageMath
    def A147597(n): return 7*(1+4^(n-2)) -(119/16)*int(n==0) -(31/4)*int(n==1) -7*int(n==2) -4*int(n==3)
[A147597(n) for n in range(1,41)] # G. C. Greubel, Oct 25 2022

Formula

From R. J. Mathar, Feb 05 2010: (Start)
a(n) = 5*a(n-1) - 4*a(n-2) for n>5.
G.f.: x*(2*x+1)*(2*x-1)*(4*x^2+2*x+1)/((4*x-1)*(1-x)). (End)
a(n) = 7*4^(n-2) + 7 for n>3. - Colin Barker, Nov 25 2016
E.g.f.: (7/16)*(16*exp(x) + exp(4*x)) -(119/16) -31*x/4 -7*x^2/2 -2*x^3/3. - G. C. Greubel, Oct 25 2022

Extensions

More terms from R. J. Mathar, Feb 05 2010
Showing 1-3 of 3 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

Content is available under The OEIS End-User License Agreement.