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.

A192809 Coefficient of x in the reduction of the polynomial (x^2 + 2)^n by x^3 -> x^2 + 2.

Original entry on oeis.org

0, 0, 2, 14, 74, 366, 1786, 8702, 42410, 206734, 1007834, 4913310, 23953034, 116774190, 569289402, 2775359806, 13530239338, 65961672910, 321571716762, 1567703857118, 7642759781962, 37259445922414, 181644634930298, 885541171698814
Offset: 0

Views

Author

Clark Kimberling, Jul 10 2011

Keywords

Comments

For discussions of polynomial reduction, see A192232 and A192744.

Crossrefs

Programs

  • GAP
    a:=[0,0,2];; for n in [4..25] do a[n]:=7*a[n-1]-12*a[n-2]+8*a[n-3]; od; Print(a); # Muniru A Asiru, Jan 02 2019
  • Magma
    m:=30; R:=PowerSeriesRing(Integers(), m); [0,0] cat Coefficients(R!( 2*x^2/(1-7*x+12*x^2-8*x^3) )); // G. C. Greubel, Jan 02 2019
    
  • Mathematica
    (See A192808.)
    LinearRecurrence[{7,-12,8}, {0,0,2}, 30] (* G. C. Greubel, Jan 02 2019 *)
  • PARI
    my(x='x+O('x^30)); concat([0,0], Vec(2*x^2/(1-7*x+12*x^2-8*x^3))) \\ G. C. Greubel, Jan 02 2019
    
  • Sage
    (2*x^2/(1-7*x+12*x^2-8*x^3)).series(x, 30).coefficients(x, sparse=False) # G. C. Greubel, Jan 02 2019
    

Formula

a(n) = 7*a(n-1) - 12*a(n-2) + 8*a(n-3).
a(n) = 2*A192811(n).
G.f.: 2*x^2/(1-7*x+12*x^2-8*x^3). - Colin Barker, Jul 26 2012

A192811 a(n) = A192809(n)/2.

Original entry on oeis.org

0, 0, 1, 7, 37, 183, 893, 4351, 21205, 103367, 503917, 2456655, 11976517, 58387095, 284644701, 1387679903, 6765119669, 32980836455, 160785858381, 783851928559, 3821379890981, 18629722961207, 90822317465149, 442770585849407
Offset: 0

Views

Author

Clark Kimberling, Jul 10 2011

Keywords

Crossrefs

Programs

  • GAP
    a:=[0,0,1];; for n in [4..25] do a[n]:=7*a[n-1]-12*a[n-2]+8*a[n-3]; od; Print(a); # Muniru A Asiru, Jan 03 2019
  • Magma
    m:=30; R:=PowerSeriesRing(Integers(), m); [0,0] cat Coefficients(R!( x^2/(1-7*x+12*x^2-8*x^3) )); // G. C. Greubel, Jan 03 2019
    
  • Mathematica
    (See A192808.)
    LinearRecurrence[{7,-12,8},{0,0,1},30] (* Harvey P. Dale, Dec 06 2018 *)
  • PARI
    my(x='x+O('x^30)); concat([0,0], Vec(x^2/(1-7*x+12*x^2-8*x^3))) \\ G. C. Greubel, Jan 03 2019
    
  • Sage
    (x^2/(1-7*x+12*x^2-8*x^3)).series(x, 30).coefficients(x, sparse=False) # G. C. Greubel, Jan 03 2019
    

Formula

a(n) = 7*a(n-1) - 12*a(n-2) + 8*a(n-3).
G.f.: x^2/(1-7*x+12*x^2-8*x^3). - Colin Barker, Jul 26 2012

Extensions

Name corrected by Colin Barker, Jul 26 2012

A192810 Coefficient of x^2 in the reduction of the polynomial (x^2 + 2)^n by x^3 -> x^2 + 2.

Original entry on oeis.org

0, 1, 5, 23, 109, 527, 2565, 12503, 60957, 297183, 1448821, 7063207, 34434061, 167870511, 818390501, 3989759863, 19450597117, 94824185471, 462280211797, 2253676033863, 10986963179245, 53562871542735, 261125950919109, 1273022903354903
Offset: 0

Views

Author

Clark Kimberling, Jul 10 2011

Keywords

Comments

For discussions of polynomial reduction, see A192232 and A192744.

Crossrefs

Programs

  • GAP
    a:=[0,1,5];; for n in [4..25] do a[n]:=7*a[n-1]-12*a[n-2]+8*a[n-3]; od; Print(a); # Muniru A Asiru, Jan 02 2019
  • Magma
    m:=30; R:=PowerSeriesRing(Integers(), m); [0] cat Coefficients(R!( x*(1-2*x)/(1-7*x+12*x^2-8*x^3) )); // G. C. Greubel, Jan 02 2019
    
  • Mathematica
    (See A192808.)
    LinearRecurrence[{7,-12,8}, {0,1,5}, 30] (* G. C. Greubel, Jan 02 2019 *)
    CoefficientList[Series[x (1-2x)/(1-7x+12x^2-8x^3),{x,0,30}],x] (* Harvey P. Dale, Aug 26 2021 *)
  • PARI
    my(x='x+O('x^30)); concat([0], Vec(x*(1-2*x)/(1-7*x+12*x^2-8*x^3) )) \\ G. C. Greubel, Jan 02 2019
    
  • Sage
    (x*(1-2*x)/(1-7*x+12*x^2-8*x^3)).series(x, 30).coefficients(x, sparse=False) # G. C. Greubel, Jan 02 2019
    

Formula

a(n) = 7*a(n-1) - 12*a(n-2) + 8*a(n-3).
G.f.: x*(1-2*x)/(1-7*x+12*x^2-8*x^3). - Colin Barker, Jul 26 2012
Showing 1-3 of 3 results.