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 1-4 of 4 results.

A138826 Concatenation of 2n-1 digits 1, n digits 0 and 2n-1 digits 1.

Original entry on oeis.org

101, 11100111, 1111100011111, 111111100001111111, 11111111100000111111111, 1111111111100000011111111111, 111111111111100000001111111111111, 11111111111111100000000111111111111111
Offset: 1

Views

Author

Omar E. Pol, Apr 06 2008

Keywords

Comments

a(n) has 5n-2 digits.
a(n) is also A147540(n) written in base 2. [Omar E. Pol, Nov 08 2008]

Examples

			n ........... a(n)
1 ........... 101
2 ......... 11100111
3 ....... 1111100011111
4 ..... 111111100001111111
5 ... 11111111100000111111111

Links

Crossrefs

Cf. A138120, A138144, A138145, A138146, A138148, A138720, A138721, A138722, A147540.

Programs

  • Mathematica
    Table[(1000^n + 10)*(100^n - 10)/900, {n, 10}] (* Paolo Xausa, Aug 08 2024 *)
  • PARI
    Vec(x*(1100000000*x^3-2000000*x^2+888910*x+101)/((x-1)*(100*x-1)*(1000*x-1)*(100000*x-1)) + O(x^100)) \\ Colin Barker, Sep 16 2013

Formula

a(n) = (10^(2n-1)-1+10^(5n-2)-10^(3n-1))/9. [R. J. Mathar, Nov 07 2008, corrected Nov 09 2008]
G.f.: x*(1100000000*x^3-2000000*x^2+888910*x+101) / ((x-1)*(100*x-1)*(1000*x-1)*(100000*x-1)). - Colin Barker, Sep 16 2013

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

A147596 a(n) is the number whose binary representation is A138145(n).

Original entry on oeis.org

1, 3, 7, 15, 31, 63, 119, 231, 455, 903, 1799, 3591, 7175, 14343, 28679, 57351, 114695, 229383, 458759, 917511, 1835015, 3670023, 7340039, 14680071, 29360135, 58720263, 117440519, 234881031, 469762055, 939524103, 1879048199, 3758096391
Offset: 1

Views

Author

Omar E. Pol, Nov 08 2008

Keywords

Links

Crossrefs

Cf. A138145, A145641, A147537, A147538, A147539.
Cf. A147540, A147590, A147595, A147597.

Programs

  • Magma
    [1,3,7,15,31] cat [7*(1+2^(n-3)): n in [6..40]]; // G. C. Greubel, Oct 25 2022
  • Mathematica
    Join[{1,3,7,15,31}, 7*(1+2^(Range[6, 40] -3))] (* G. C. Greubel, Oct 25 2022 *)
  • PARI
    Vec(-x*(2*x^2-1)*(4*x^4+2*x^2+1)/((x-1)*(2*x-1)) + O(x^100)) \\ Colin Barker, Sep 15 2013
  • SageMath
    def A147596(n): return 7*(1+2^(n-3)) -(1/8)*(63*int(n==0) +62*int(n==1) +60*int(n ==2)) -(7*int(n==3) +6*int(n==4) +4*int(n==5))
[A147596(n) for n in range(1,40)] # G. C. Greubel, Oct 25 2022

Formula

a(n) = 7*(2^(n-3) + 1) if n >= 6. - Hagen von Eitzen, Jun 02 2009
From Colin Barker, Sep 15 2013: (Start)
a(n) = 3*a(n-1) - 2*a(n-2), for n >= 8.
G.f.: x*(1-2*x^2)*(1+2*x^2+4*x^4) / ((1-x)*(1-2*x)). (End)
E.g.f.: (7/8)*(8*exp(x) + exp(2*x)) - (1/8)*(63 + 62*x + 30*x^2) - 7*x^3/6 - x^4/4 - x^5/30. - G. C. Greubel, Oct 25 2022

Extensions

More terms from Hagen von Eitzen, Jun 02 2009

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 1-4 of 4 results.

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