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.

A275858 a(n) = floor(c*r*a(n-1)) - floor(d*s*a(n-2)), where r = (1+sqrt(5))/2, s = r/(r-1), c = 1, d = 1, a(0) = 1, a(1) = 1.

Original entry on oeis.org

1, 1, -1, -4, -4, 4, 17, 17, -17, -72, -72, 72, 305, 305, -305, -1292, -1292, 1292, 5473, 5473, -5473, -23184, -23184, 23184, 98209, 98209, -98209, -416020, -416020, 416020, 1762289, 1762289, -1762289, -7465176, -7465176, 7465176, 31622993, 31622993
Offset: 0

Views

Author

Clark Kimberling, Aug 12 2016

Keywords

Links

Crossrefs

Cf. A275856, A275857, A275859, A275860, A275861.

Programs

  • Magma
    Q:=Rationals(); R:=PowerSeriesRing(Q, 40); Coefficients(R!(1/(1-x+2*x^2+x^3+x^4))) // G. C. Greubel, Feb 08 2018
  • Mathematica
    c = 1; d = 1; z = 40;
r = (c + Sqrt[c^2 + 4 d])/2; s = r/(r - 1); a[0] = 1; a[1] = 1;
a[n_] := a[n] = Floor[c*s*a[n - 1]] + Floor[d*r*a[n - 2]];
t = Table[a[n], {n, 0, z}] (* A275856 *)
CoefficientList[Series[1/(1-x+2*x^2+x^3+x^4), {x,0, 50}], x] (* G. C. Greubel, Feb 08 2018 *)
  • PARI
    x='x+O('x^30); Vec(1/(1-x+2*x^2+x^3+x^4)) \\ G. C. Greubel, Feb 08 2018

Formula

a(n) = floor(r*a(n-1)) - floor(s*a(n-2)), where r = (1+sqrt(5))/2, s = r/(r-1).
G.f.: 1/(1 - x + 2*x^2 + x^3 + x^4).

A275857 a(n) = floor(c*s*a(n-1)) + floor(d*r*a(n-2)), where r = (1+sqrt(5))/2, s = r/(r-1), c = 1, d = 1, a(0) = 1, a(1) = 2.

Original entry on oeis.org

1, 2, 6, 18, 56, 175, 548, 1717, 5381, 16865, 52859, 165674, 519267, 1627524, 5101104, 15988252, 50111546, 157063265, 492279150, 1542937247, 4835986551, 15157302067, 47507122597, 148900291588, 466694163381, 1462746914806, 4584648158602, 14369538930774
Offset: 0

Views

Author

Clark Kimberling, Aug 11 2016

Keywords

Links

Crossrefs

Cf. A275856, A275858-A275861.

Programs

  • Mathematica
    c = 1; d = 1; z = 40;
r = (c + Sqrt[c^2 + 4 d])/2; s = r/(r - 1); a[0] = 1; a[1] = 1;
a[n_] := a[n] = Floor[c*s*a[n - 1]] + Floor[d*r*a[n - 2]];
t = Table[a[n], {n, 0, z}] (* A275856 *)

Formula

a(n) = floor(s*a(n-1)) + floor(r*a(n-2)), where r = (1+sqrt(5))/2, s = r/(r-1).
G.f.: (1 - 2 x + x^2 - x^3)/(1 - 4 x + 3 x^2 - x^3 + x^5).

A275861 a(n) = floor(c*s*a(n-1)) + floor(d*r*a(n-2)), where r = 2 + sqrt(5), s = r/(r-1), c = 4, d = 1, a(0) = 1, a(1) = 1.

Original entry on oeis.org

1, 1, 9, 51, 305, 1813, 10784, 64144, 381543, 2269503, 13499513, 80298135, 477631347, 2841058559, 16899254596, 100520563016, 597918892325, 3556555903317, 21155193548465, 125835844069155, 748499871500621, 4452245397810113, 26482955892270832
Offset: 0

Views

Author

Clark Kimberling, Aug 12 2016

Keywords

Links

Crossrefs

Cf. A275856, A275857, A275858, A275859, A275860.

Programs

  • Magma
    Q:=Rationals(); R:=PowerSeriesRing(Q, 40); Coefficients(R!((1-6*x+9*x^2-10*x^3+9*x^4-4*x^5)/(1-7*x+7*x^2 -5*x^3+3*x^4+ 3*x^5- 3*x^6+x^7))) // G. C. Greubel, Feb 08 2018
  • Mathematica
    c = 4; d = 1; z = 40;
r = (c + Sqrt[c^2 + 4 d])/2; s = r/(r - 1); a[0] = 1; a[1] = 1;
a[n_] := a[n] = Floor[c*s*a[n - 1]] + Floor[d*r*a[n - 2]];
t = Table[a[n], {n, 0, z}]
CoefficientList[Series[(1-6*x+9*x^2-10*x^3+9*x^4-4*x^5)/(1-7*x+7*x^2 -5*x^3+3*x^4+3*x^5-3*x^6+x^7), {x,0, 50}], x] (* G. C. Greubel, Feb 08 2018 *)
LinearRecurrence[{7,-7,5,-3,-3,3,-1},{1,1,9,51,305,1813,10784},40] (* Harvey P. Dale, Dec 21 2018 *)
  • PARI
    x='x+O('x^30); Vec((1-6*x+9*x^2-10*x^3+9*x^4-4*x^5)/(1-7*x+7*x^2 -5*x^3+3*x^4+3*x^5-3*x^6+x^7)) \\ G. C. Greubel, Feb 08 2018

Formula

a(n) = floor(c*s*a(n-1)) + floor(d*r*a(n-2)), where r = 2 + sqrt(5), s = r/(r-1), c = 4, d = 1, a(0) = 1, a(1) = 1.
G.f.: (1 -6*x +9*x^2 -10*x^3 +9*x^4 -4*x^5)/(1 -7*x +7*x^2 -5*x^3 +3*x^4 +3*x^5 -3*x^6 +x^7).

A275859 a(n) = floor(c*s*a(n-1)) + floor(d*r*a(n-2)), where r = 1 + sqrt(2), s = r/(r-1), c = 2, d = 1, a(0) = 1, a(1) = 1.

Original entry on oeis.org

1, 1, 5, 19, 76, 304, 1220, 4898, 19667, 78971, 317103, 1273309, 5112902, 20530578, 82439414, 331030964, 1329236757, 5337477605, 21432349833, 86060430295, 345570957936, 1387621309348, 5571917587224, 22373730779190, 89840494074695, 360749597608127
Offset: 0

Views

Author

Clark Kimberling, Aug 12 2016

Keywords

Links

Crossrefs

Cf. A275856, A275857, A275858, A275860, A275861.

Programs

  • Magma
    Q:=Rationals(); R:=PowerSeriesRing(Q, 40); Coefficients(R!((1-4*x+4*x^2-2*x^3)/(1-5*x+4*x^2-x^4+x^5))) // G. C. Greubel, Feb 08 2018
  • Mathematica
    c = 2; d = 1; z = 40;
r = (c + Sqrt[c^2 + 4 d])/2; s = r/(r - 1); a[0] = 1; a[1] = 1;
a[n_] := a[n] = Floor[c*s*a[n - 1]] + Floor[d*r*a[n - 2]];
t = Table[a[n], {n, 0, z}]
CoefficientList[Series[(1-4*x+4*x^2-2*x^3)/(1-5*x+4*x^2-x^4+x^5), {x,0, 50}], x] (* G. C. Greubel, Feb 08 2018 *)
LinearRecurrence[{5,-4,0,1,-1},{1,1,5,19,76},30] (* Harvey P. Dale, Apr 23 2019 *)
  • PARI
    x='x+O('x^30); Vec((1-4*x+4*x^2-2*x^3)/(1-5*x+4*x^2-x^4+x^5)) \\ G. C. Greubel, Feb 08 2018

Formula

a(n) = floor(s*a(n-1)) + floor(r*a(n-2)), where r = 1 + sqrt(2), s = r/(r-1).
G.f.: (1 - 4*x + 4*x^2 - 2*x^3)/(1 - 5*x + 4*x^2 - x^4 + x^5).

A275860 a(n) = floor(c*s*a(n-1)) + floor(d*r*a(n-2)), where r = (3 + sqrt(13))/2, s = r/(r-1), c = 3, d = 1, a(0) = 1, a(1) = 1.

Original entry on oeis.org

1, 1, 7, 33, 164, 813, 4039, 20063, 99665, 495099, 2459470, 12217747, 60693301, 301502133, 1497752387, 7440286381, 36960623072, 183606865105, 912091791531, 4530938620963, 22508046862781, 111811749387479, 555439900107962, 2759222392297991, 13706808258965257
Offset: 0

Views

Author

Clark Kimberling, Aug 12 2016

Keywords

Links

Crossrefs

Cf. A098316, A275856, A275857, A275858, A275859, A275861.

Programs

  • Magma
    Q:=Rationals(); R:=PowerSeriesRing(Q, 40); Coefficients(R!((1-4*x+2*x^2-2*x^3+3*x^4-3*x^5+x^6)/(1-5*x+4*x^4-x^6+x^7))); // G. C. Greubel, Feb 08 2018
  • Mathematica
    c = 3; d = 1; z = 40;
r = (c + Sqrt[c^2 + 4 d])/2; s = r/(r - 1); a[0] = 1; a[1] = 1;
a[n_] := a[n] = Floor[c*s*a[n - 1]] + Floor[d*r*a[n - 2]];
t = Table[a[n], {n, 0, z}]
CoefficientList[Series[(1-4*x+2*x^2-2*x^3+3*x^4-3*x^5+x^6)/(1-5*x+4*x^4-x^6+x^7), {x,0, 50}], x] (* G. C. Greubel, Feb 08 2018 *)
LinearRecurrence[{5,0,0,-4,0,1,-1},{1,1,7,33,164,813,4039},30] (* Harvey P. Dale, Sep 17 2024 *)
  • PARI
    my(x='x+O('x^30)); Vec((1-4*x+2*x^2-2*x^3+3*x^4-3*x^5+x^6)/(1-5*x +4*x^4-x^6+x^7)) \\ G. C. Greubel, Feb 08 2018

Formula

a(n) = floor(c*s*a(n-1)) + floor(d*r*a(n-2)), where r = (3 + sqrt(13))/2 (A098316), s = r/(r-1), c = 3, d = 1, a(0) = 1, a(1) = 1.
G.f.: (1 -4*x +2*x^2 -2*x^3 +3*x^4 -3*x^5 +x^6)/(1 -5*x +4*x^4 -x^6 +x^7).
Showing 1-5 of 5 results.

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