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.

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).

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).

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 1-3 of 3 results.

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