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 1-8 of 8 results.

A134654 Duplicate of A038521.

Original entry on oeis.org

0, 0, 2, 1, 4, 10, 12, 36, 64, 120, 272, 496, 1024, 2080, 4032, 8256, 16384, 32640, 65792, 130816, 262144, 524800, 1047552, 2098176, 4194304
Offset: 0

Views

Author

Keywords

A038518 Number of elements of GF(2^n) with trace 0 and subtrace 0.

Original entry on oeis.org

0, 1, 1, 1, 6, 6, 16, 36, 56, 136, 256, 496, 1056, 2016, 4096, 8256, 16256, 32896, 65536, 130816, 262656, 523776, 1048576, 2098176, 4192256, 8390656, 16777216, 33550336, 67117056, 134209536, 268435456, 536887296, 1073709056, 2147516416
Offset: 0

Views

Author

Frank Ruskey

Keywords

Links

Crossrefs

Cf. A038503, A038505.
Cf. A038519, A038520, A038521.

Programs

  • Maple
    0,seq(1/4*2^k-1/4*(-1-I)^k-1/4*(-1+I)^k,k=1..40); seq(coeff(convert(series((-x^3+x^2+x)/((1-2*x)*(1+2*x+2*x^2)),x,50),polynom),x,i),i=0..40); # C. Ronaldo (aga_new_ac(AT)hotmail.com), Dec 16 2004
  • Mathematica
    LinearRecurrence[{0,2,4},{0,1,1,1},40] (* Harvey P. Dale, Mar 31 2020 *)
  • PARI
    concat(0, Vec(x*(1 + x - x^2) / ((1 - 2*x)*(1 + 2*x + 2*x^2)) + O(x^40))) \\ Colin Barker, Aug 02 2019

Formula

C(n, r+0)+C(n, r+4)+C(n, r+8)+... where r = 0 if n odd, r = 2 if n even.
G.f.: (-x^3+x^2+x)/[(1-2x)(1+2x+2x^2)].
a(0)=0; a(n) = ( 2^n - (-1-i)^n - (-1+i)^n )/4, i=sqrt(-1). - C. Ronaldo (aga_new_ac(AT)hotmail.com), Dec 16 2004
a(n) = 2*a(n-2) + 4*a(n-3) for n>3. - Colin Barker, Aug 02 2019

A038519 Number of elements of GF(2^n) with trace 0 and subtrace 1.

Original entry on oeis.org

1, 0, 1, 3, 2, 10, 16, 28, 72, 120, 256, 528, 992, 2080, 4096, 8128, 16512, 32640, 65536, 131328, 261632, 524800, 1048576, 2096128, 4196352, 8386560, 16777216, 33558528, 67100672, 134225920, 268435456, 536854528, 1073774592
Offset: 0

Views

Author

Frank Ruskey

Keywords

Links

Crossrefs

Cf. A038503, A038505, A038518, A038520, A038521.

Programs

  • Magma
    I:=[1,0,1,3]; [m le 4 select I[m] else 2*Self(m-2)+4*Self(m-3):m in [1..33]]; // Marius A. Burtea, Aug 02 2019
  • PARI
    Vec((1 - x^2 - x^3) / ((1 - 2*x)*(1 + 2*x + 2*x^2)) + O(x^40)) \\ Colin Barker, Aug 02 2019

Formula

a(n) = C(n, r+0) + C(n, r+4) + C(n, r+8) + ... where r = 2 if n odd, r = 0 if n even.
From Colin Barker, Aug 02 2019: (Start)
G.f.: (1 - x^2 - x^3) / ((1 - 2*x)*(1 + 2*x + 2*x^2)). - Creighton Dement, Apr 29 2005, corrected by Colin Barker, Aug 02 2019
a(n) = ((-1-i)^n + (-1+i)^n + 2^n) / 4 for n>0.
a(n) = 2*a(n-2) + 4*a(n-3) for n>3.
(End)

A038520 Number of elements of GF(2^n) with trace 1 and subtrace 0.

Original entry on oeis.org

0, 1, 0, 3, 4, 6, 20, 28, 64, 136, 240, 528, 1024, 2016, 4160, 8128, 16384, 32896, 65280, 131328, 262144, 523776, 1049600, 2096128, 4194304, 8390656, 16773120, 33558528, 67108864, 134209536, 268451840, 536854528, 1073741824
Offset: 0

Views

Author

Frank Ruskey

Keywords

Links

Crossrefs

Cf. A038504, A000749, A038518, A038519, A038521.

Programs

  • Mathematica
    LinearRecurrence[{0,2,4},{0,1,0,3},40] (* Harvey P. Dale, Oct 15 2017 *)
  • PARI
    concat(0, Vec(x*(1 + x^2) / ((1 - 2*x)*(1 + 2*x + 2*x^2)) + O(x^40))) \\ Colin Barker, Aug 02 2019

Formula

a(n) = C(n, r+0)+C(n, r+4)+C(n, r+8)+... where r = 1 if n odd, r = 3 if n even.
a(n) = 2*a(n-2) + 4*a(n-3), n > 3. - Paul Curtz, Feb 06 2008
From Colin Barker, Aug 02 2019: (Start)
G.f.: x*(1 + x^2) / ((1 - 2*x)*(1 + 2*x + 2*x^2)).
a(n) = (2^n + i*((-1-i)^n - (-1+i)^n)) / 4 for n>0, where i=sqrt(-1).
(End)

A134142 List of quadruples: 2*(-4)^n, -3*(-4)^n, 2*(-4^n), 2*(-4)^n, n >= 0.

Original entry on oeis.org

2, -3, 2, 2, -8, 12, -8, -8, 32, -48, 32, 32, -128, 192, -128, -128, 512, -768, 512, 512, -2048, 3072, -2048, -2048, 8192, -12288, 8192, 8192, -32768, 49152, -32768, -32768, 131072, -196608, 131072, 131072, -524288, 786432, -524288, -524288, 2097152, -3145728, 2097152, 2097152, -8388608
Offset: 0

Views

Author

Paul Curtz, Jan 29 2008

Keywords

Links

Crossrefs

Cf. A081147, A038521.

Programs

  • Maple
    A134142 := proc(n) (-4)^floor(n/4)*op(1+(n mod 4), [2,-3,2,2]) ; end: for n from 0 to 80 do printf("%d ",A134142(n)) ; od: # R. J. Mathar, Feb 05 2008

Formula

O.g.f.: (x+2)/(2*x^2+2*x+1). a(n) = 2*A108520(n)+A108520(n-1). - R. J. Mathar, Feb 05 2008
a(n) = (1 - I/2)*(-1 - I)^n + (1 + I/2)*(-1 + I)^n, n>=0. - Taras Goy, Apr 20 2019
a(n) = -2*a(n-1)-2*a(n-2) for n > 1. - Chai Wah Wu, May 19 2025
E.g.f.: exp(-x)*(2*cos(x) - sin(x)). - Stefano Spezia, May 19 2025

Extensions

More terms from R. J. Mathar, Feb 05 2008

A134136 a(n) = 2*a(n-2) + 4*a(n-3), with initial terms 0, 1, 1.

Original entry on oeis.org

0, 1, 1, 2, 6, 8, 20, 40, 72, 160, 304, 608, 1248, 2432, 4928, 9856, 19584, 39424, 78592, 157184, 314880, 628736, 1258496, 2516992, 5031936, 10067968, 20131840, 40263680, 80535552, 161054720, 322125824, 644251648, 1288470528, 2577006592, 5153947648, 10307895296
Offset: 0

Views

Author

Paul Curtz, Jan 29 2008

Keywords

Links

Crossrefs

Cf. A038521.

Programs

  • Magma
    [n le 3 select Floor(n/2) else 2*Self(n-2)+4*Self(n-3): n in [1..40]]; // Vincenzo Librandi, May 28 2015
  • Maple
    f:= gfun:-rectoproc({a(n)=2*a(n-2)+4*a(n-3), a(0)=0,a(1)=1,a(2)=1},a(n),remember):
seq(f(n),n=0..100); # Robert Israel, May 27 2015
  • Mathematica
    Nest[Append[#, 2 #[[-2]] + 4 #[[-3]]] &, {0, 1, 1}, 15] (* Ivan Neretin, May 27 2015 *)
CoefficientList[Series[x (1 + x)/((1 - 2 x) (2 x^2 + 2 x + 1)), {x, 0, 40}], x] (* Vincenzo Librandi, May 28 2015 *)

Formula

a(n) = (6*2^n - (3+i)*(-1+i)^n - (3-i)*(-1-i)^n)/20. - Ivan Neretin, May 27 2015
G.f.: (x^2+x)/(1-2*x^2-4*x^3). - Robert Israel, May 27 2015

Extensions

More terms from Robert Israel, May 27 2015

A134812 a(n) = 2a(n-2) + 4a(n-3), n >= 3.

Original entry on oeis.org

0, 1, -1, 1, 2, -2, 8, 4, 8, 40, 32, 112, 224, 352, 896, 1600, 3200, 6784, 12800, 26368, 52736, 103936, 210944, 418816, 837632, 1681408, 3350528, 6713344, 13426688, 26828800, 53706752, 107364352, 214728704, 429555712, 858914816, 1718026240, 3436052480, 6871711744, 13744209920
Offset: 0

Views

Author

Paul Curtz, Jan 28 2008

Keywords

Links

Crossrefs

Cf. A038521.

Programs

  • Mathematica
    CoefficientList[Series[x*(-1 + x + x^2) / ( (2*x-1)*(2*x^2 + 2*x + 1) ),{x,0,38}],x] (* James C. McMahon, Apr 11 2025 *)

Formula

G.f.: x*(-1 + x + x^2) / ( (2*x-1)*(2*x^2 + 2*x + 1) ). - R. J. Mathar, Aug 11 2012
20*a(n) = 2^n - 6*A078069(n), n>0. - R. J. Mathar, Aug 11 2012

Extensions

More terms from Joshua Zucker, Feb 23 2008

A134068 a(n) = 2*a(n-2) + 4*a(n-3), with initial terms 0, 3, 3.

Original entry on oeis.org

0, 3, 3, 6, 18, 24, 60, 120, 216, 480, 912, 1824, 3744, 7296, 14784, 29568, 58752, 118272, 235776, 471552, 944640, 1886208, 3775488, 7550976, 15095808, 30203904, 60395520, 120791040, 241606656, 483164160, 966377472, 1932754944, 3865411584, 7731019776, 15461842944
Offset: 0

Views

Author

Paul Curtz, Jan 29 2008

Keywords

Links

Crossrefs

Cf. A134136, A038521.

Programs

  • Magma
    a:=[0,3,3]; [n le 3 select a[n] else 2*Self(n-2) + 4*Self(n-3):n in [1..35]]; // Marius A. Burtea, Jan 03 2020
  • Magma
    R:=PowerSeriesRing(Integers(), 35); [0] cat Coefficients(R!( 3*x*(1 + x)/((1 - 2*x)*(1 + 2*x + 2*x^2)))); // Marius A. Burtea, Jan 03 2020
  • PARI
    concat([0], Vec(3*(1 + x)/((1 - 2*x)*(1 + 2*x + 2*x^2)) + O(x^40))) \\ Andrew Howroyd, Jan 03 2020

Formula

From Andrew Howroyd, Jan 03 2020: (Start)
G.f.: 3*x*(1 + x)/((1 - 2*x)*(1 + 2*x + 2*x^2)).
a(n) = 3*A134136(n). (End)

Extensions

a(12) corrected and terms a(13) and beyond from Andrew Howroyd, Jan 03 2020
Showing 1-8 of 8 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.