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 1-5 of 5 results.

A054145 Row sums of array T as in A054144.

Original entry on oeis.org

0, 2, 12, 58, 256, 1072, 4336, 17112, 66304, 253280, 956608, 3579680, 13292544, 49039360, 179912448, 656874368, 2388205568, 8650598912, 31231020032, 112419973632, 403596148736, 1445463642112, 5165581660160, 18423238924288
Offset: 0

Views

Author

Clark Kimberling, Mar 18 2000

Keywords

Programs

  • GAP
    a:=[0,2,12,58];; for n in [5..30] do a[n]:=8*a[n-1]-20*a[n-2] +16*a[n-3]-4*a[n-4]; od; a; # G. C. Greubel, Jul 31 2019
  • Magma
    R:=PowerSeriesRing(Integers(), 30); [0] cat Coefficients(R!( 2*x*(1-x)^2/(1-4*x+2*x^2)^2 )); // G. C. Greubel, Jul 31 2019
    
  • Mathematica
    LinearRecurrence[{8,-20,16,-4}, {0,2,12,58}, 30] (* G. C. Greubel, Jul 31 2019 *)
  • PARI
    my(x='x+O('x^30)); concat([0], Vec(2*x*(1-x)^2/(1-4*x+2*x^2)^2)) \\ G. C. Greubel, Jul 31 2019
    
  • Sage
    (2*x*(1-x)^2/(1-4*x+2*x^2)^2).series(x, 30).coefficients(x, sparse=False) # G. C. Greubel, Jul 31 2019
    

Formula

G.f.: 2*x*(1 - x)^2/(1 - 4*x + 2*x^2)^2.
a(n) = ((n-2)*((2 + sqrt(2))^n + (2 - sqrt(2))^n) + sqrt(2)*((2 + sqrt(2))^n - (2 - sqrt(2))^n))/8. - G. C. Greubel, Jul 31 2019

A054147 a(n) = T(2n,n), array T as in A054144.

Original entry on oeis.org

0, 3, 21, 108, 492, 2100, 8604, 34272, 133728, 513648, 1948560, 7318080, 27256896, 100815936, 370684608, 1355996160, 4938304512, 17914202880, 64760732928, 233390693376, 838784916480, 3006980379648, 10755352869888
Offset: 0

Views

Author

Clark Kimberling, Mar 18 2000

Keywords

Crossrefs

Cf. A054144.

Programs

  • GAP
    a:=[0,3,21,108];; for n in [5..30] do a[n]:=8*a[n-1]-20*a[n-2] +16*a[n-3]-4*a[n-4]; od; a; # G. C. Greubel, Jul 31 2019
  • Magma
    R:=PowerSeriesRing(Integers(), 30); [0] cat Coefficients(R!( 3*x*(1-x)/(1-4*x+2*x^2)^2 )); // G. C. Greubel, Jul 31 2019
    
  • Mathematica
    LinearRecurrence[{8,-20,16,-4}, {0,3,21,108}, 30] (* G. C. Greubel, Jul 31 2019 *)
  • PARI
    my(x='x+O('x^30)); concat([0], Vec(3*x*(1-x)/(1-4*x+2*x^2)^2)) \\ G. C. Greubel, Jul 31 2019
    
  • Sage
    (3*x*(1-x)/(1-4*x+2*x^2)^2).series(x, 30).coefficients(x, sparse=False) # G. C. Greubel, Jul 31 2019
    

Formula

G.f.: 3*x*(1-x)/(1-4*x+2*x^2)^2.
From Colin Barker, Aug 01 2019: (Start)
a(n) = 8*a(n-1) - 20*a(n-2) + 16*a(n-3) - 4*a(n-4) for n>3.
a(n) = 3*((-(2-sqrt(2))^n*(-1+sqrt(2)) + (1+sqrt(2))*(2+sqrt(2))^n)*n) / 8.
(End)

A054148 T(2n+1,n), array T as in A054144.

Original entry on oeis.org

1, 5, 28, 132, 574, 2380, 9560, 37536, 144872, 551696, 2078464, 7761600, 28771168, 105985984, 388336256, 1416262656, 5144067200, 18616720640, 67159278592, 241579840512, 866744413696, 3102440074240, 11081272653824
Offset: 0

Views

Author

Clark Kimberling, Mar 18 2000

Keywords

A054149 T(2n-1,n), array T as in A054144.

Original entry on oeis.org

1, 14, 84, 410, 1820, 7648, 31008, 122584, 475600, 1818656, 6874560, 25742624, 95645888, 353032960, 1295729664, 4732541824, 17211685120, 62362187264, 225201546240, 810825419264, 2911520685056, 10429433085952
Offset: 1

Views

Author

Clark Kimberling, Mar 18 2000

Keywords

Formula

Conjectures from Colin Barker, Feb 11 2015: (Start)
a(n) = 8*a(n-1)-20*a(n-2)+16*a(n-3)-4*a(n-4).
G.f.: x*(2*x^3-8*x^2+6*x+1) / (2*x^2-4*x+1)^2.
(End)

A054146 a(n) = A054145(n)/2.

Original entry on oeis.org

0, 1, 6, 29, 128, 536, 2168, 8556, 33152, 126640, 478304, 1789840, 6646272, 24519680, 89956224, 328437184, 1194102784, 4325299456, 15615510016, 56209986816, 201798074368, 722731821056, 2582790830080, 9211619462144
Offset: 0

Views

Author

Clark Kimberling, Mar 18 2000

Keywords

Crossrefs

Programs

  • GAP
    a:=[0,1,6,29];; for n in [5..30] do a[n]:=8*a[n-1]-20*a[n-2] +16*a[n-3]-4*a[n-4]; od; a; # G. C. Greubel, Aug 01 2019
  • Magma
    R:=PowerSeriesRing(Integers(), 30); [0] cat Coefficients(R!( x*(1-x)^2/(1-4*x+2*x^2)^2 )); // G. C. Greubel, Aug 01 2019
    
  • Mathematica
    LinearRecurrence[{8,-20,16,-4}, {0,1,6,29}, 30] (* G. C. Greubel, Aug 01 2019 *)
  • PARI
    my(x='x+O('x^30)); concat([0], Vec(x*(1-x)^2/(1-4*x+2*x^2)^2)) \\ G. C. Greubel, Aug 01 2019
    
  • Sage
    (x*(1-x)^2/(1-4*x+2*x^2)^2).series(x, 30).coefficients(x, sparse=False) # G. C. Greubel, Aug 01 2019
    

Formula

From G. C. Greubel, Aug 01 2019: (Start)
a(n) = ((n-2)*((2 + sqrt(2))^n + (2 - sqrt(2))^n) + sqrt(2)*((2 + sqrt(2))^n - (2 - sqrt(2))^n))/16.
G.f.: x*(1 - x)^2/(1 - 4*x + 2*x^2)^2. (End)
Showing 1-5 of 5 results.