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.

A192777 Constant term in the reduction of the n-th Fibonacci polynomial by x^3->x^2+3x+1. See Comments.

Original entry on oeis.org

1, 0, 1, 1, 2, 8, 14, 55, 121, 392, 989, 2912, 7797, 22104, 60553, 169289, 467622, 1300888, 3603914, 10008543, 27755249, 77034176, 213702153, 593005504, 1645265209, 4565154816, 12666317073, 35144684065, 97512548090, 270561677224
Offset: 1

Views

Author

Clark Kimberling, Jul 09 2011

Keywords

Comments

For discussions of polynomial reduction, see A192232 and A192744.

Examples

			The first five polynomials p(n,x) and their reductions are as follows:
F1(x)=1 -> 1
F2(x)=x -> x
F3(x)=x^2+1 -> x^2+1
F4(x)=x^3+2x -> x^2+5x+1
F5(x)=x^4+3x^2+1 -> 7x^2+4x+2, so that
A192777=(1,0,1,1,2,...), A192778=(0,1,0,5,4,...), A192779=(0,0,1,1,7,...)

Links

Crossrefs

Cf. A192744, A192232, A192616, A192772, A192778, A192779.

Programs

  • Mathematica
    q = x^3; s = x^2 + 3 x + 1; z = 40;
p[n_, x_] := Fibonacci[n, x];
Table[Expand[p[n, x]], {n, 1, 7}]
reduce[{p1_, q_, s_, x_}] :=
FixedPoint[(s PolynomialQuotient @@ #1 +
       PolynomialRemainder @@ #1 &)[{#1, q, x}] &, p1]
t = Table[reduce[{p[n, x], q, s, x}], {n, 1, z}];
u1 = Table[Coefficient[Part[t, n], x, 0], {n, 1, z}]
  (* A192777 *)
u2 = Table[Coefficient[Part[t, n], x, 1], {n, 1, z}]
  (* A192778 *)
u3 = Table[Coefficient[Part[t, n], x, 2], {n, 1, z}]
  (* A192779 *)

Formula

a(n)=a(n-1)+6*a(n-2)-a(n-3)-6*a(n-4)+a(n-5)+a(n-6).
G.f.: -x*(1-5*x^2+x^4-x+x^3) / ( (x^2-x-1)*(x^4+2*x^3-3*x^2-2*x+1) ). - R. J. Mathar, May 06 2014

A192778 Coefficient of x in the reduction of the n-th Fibonacci polynomial by x^3->x^2+3x+1.

Original entry on oeis.org

0, 1, 0, 5, 4, 28, 48, 183, 424, 1315, 3420, 9864, 26756, 75237, 207128, 577345, 1597624, 4439764, 12307388, 34166643, 94769936, 262998791, 729644824, 2024614928, 5617339496, 15586328073, 43245649904, 119991232893, 332929027020
Offset: 1

Views

Author

Clark Kimberling, Jul 09 2011

Keywords

Comments

For discussions of polynomial reduction, see A192232 and A192744.

Examples

			The first five polynomials p(n,x) and their reductions:
F1(x)=1 -> 1
F2(x)=x -> x
F3(x)=x^2+1 -> x^2+1
F4(x)=x^3+2x -> x^2+5x+1
F5(x)=x^4+3x^2+1 -> 7x^2+4x+2, so that
A192777=(1,0,1,1,2,...), A192778=(0,1,0,5,4,...), A192779=(0,0,1,1,7,...)

Links

Crossrefs

Cf. A192744, A192232, A192616, A192772, A192777, A192779.

Formula

a(n) = a(n-1)+6*a(n-2)-a(n-3)-6*a(n-4)+a(n-5)+a(n-6).
G.f.: x^2*(x^2+x-1)/((x^2-x-1)*(x^4+2*x^3-3*x^2-2*x+1)). [Colin Barker, Nov 23 2012]

A192780 Constant term in the reduction of the n-th Fibonacci polynomial by x^3->x^2+1. See Comments.

Original entry on oeis.org

1, 0, 1, 1, 2, 5, 8, 19, 34, 71, 137, 272, 537, 1056, 2089, 4112, 8121, 16009, 31586, 62301, 122888, 242411, 478146, 943183, 1860433, 3669792, 7238769, 14278720, 28165265, 55556896, 109587889, 216165713, 426394178, 841076725, 1659052040
Offset: 1

Views

Author

Clark Kimberling, Jul 09 2011

Keywords

Comments

For discussions of polynomial reduction, see A192232 and A192744.

Examples

			The first five polynomials p(n,x) and their reductions:
F1(x)=1 -> 1
F2(x)=x -> x
F3(x)=x^2+1 -> x^2+1
F4(x)=x^3+2x -> x^2+2x+1
F5(x)=x^4+3x^2+1 -> 4x^2+1x+2, so that
A192777=(1,0,1,1,2,...), A192778=(0,1,0,2,1,...), A192779=(0,0,1,1,4,...)

Links

Crossrefs

Cf. A192744, A192232, A192616, A192781, A192782.

Programs

  • Mathematica
    q = x^3; s = x^2 + 1; z = 40;
p[n_, x_] := Fibonacci[n, x];
Table[Expand[p[n, x]], {n, 1, 7}]
reduce[{p1_, q_, s_, x_}] :=
FixedPoint[(s PolynomialQuotient @@ #1 +
       PolynomialRemainder @@ #1 &)[{#1, q, x}] &, p1]
t = Table[reduce[{p[n, x], q, s, x}], {n, 1, z}];
u1 = Table[Coefficient[Part[t, n], x, 0], {n, 1, z}]
  (* A192780 *)
u2 = Table[Coefficient[Part[t, n], x, 1], {n, 1, z}]
  (* A192781 *)
u3 = Table[Coefficient[Part[t, n], x, 2], {n, 1, z}]
  (* A192782 *)
LinearRecurrence[{1,3,-1,-3,1,1},{1,0,1,1,2,5},40] (* Harvey P. Dale, Nov 07 2021 *)

Formula

a(n)=a(n-1)+3*a(n-2)-a(n-3)-3*a(n-4)+a(n-5)+a(n-6).
G.f.: -x*(x-1)*(1+x)*(x^2+x-1) / ( -1+x+3*x^2-x^3-3*x^4+x^5+x^6 ). - R. J. Mathar, May 06 2014

A192781 Coefficient of x in the reduction of the n-th Fibonacci polynomial by x^3->x^2+1.

Original entry on oeis.org

0, 1, 0, 2, 1, 4, 6, 12, 25, 46, 96, 183, 368, 720, 1424, 2809, 5536, 10930, 21545, 42516, 83846, 165404, 326257, 643550, 1269440, 2503983, 4939232, 9742752, 19217952, 37908017, 74774848, 147495906, 290940561, 573890084, 1132017286, 2232942124
Offset: 1

Views

Author

Clark Kimberling, Jul 09 2011

Keywords

Comments

For discussions of polynomial reduction, see A192232 and A192744.

Examples

			The first five polynomials p(n,x) and their reductions:
F1(x)=1 -> 1
F2(x)=x -> x
F3(x)=x^2+1 -> x^2+1
F4(x)=x^3+2x -> x^2+2x+1
F5(x)=x^4+3x^2+1 -> 4x^2+1x+2, so that
A192777=(1,0,1,1,2,...), A192778=(0,1,0,2,1,...), A192779=(0,0,1,1,4,...)

Links

Crossrefs

Cf. A192744, A192232, A192616, A192780, A192782.

Programs

  • Mathematica
    q = x^3; s = x^2 + 1; z = 40;
p[n_, x_] := Fibonacci[n, x];
Table[Expand[p[n, x]], {n, 1, 7}]
reduce[{p1_, q_, s_, x_}] :=
FixedPoint[(s PolynomialQuotient @@ #1 +
       PolynomialRemainder @@ #1 &)[{#1, q, x}] &, p1]
t = Table[reduce[{p[n, x], q, s, x}], {n, 1, z}];
u1 = Table[Coefficient[Part[t, n], x, 0], {n, 1, z}]
  (* A192780 *)
u2 = Table[Coefficient[Part[t, n], x, 1], {n, 1, z}]
  (* A192781 *)
u3 = Table[Coefficient[Part[t, n], x, 2], {n, 1, z}]
  (* A192782 *)

Formula

a(n) = a(n-1)+3*a(n-2)-a(n-3)-3*a(n-4)+a(n-5)+a(n-6).
G.f.: x^2*(x^2+x-1)/(x^6+x^5-3*x^4-x^3+3*x^2+x-1). [Colin Barker, Nov 23 2012]

A192782 Coefficient of x in the reduction of the n-th Fibonacci polynomial by x^3->x^2+1.

Original entry on oeis.org

0, 0, 1, 1, 4, 6, 14, 26, 52, 103, 201, 400, 784, 1552, 3056, 6032, 11897, 23465, 46292, 91302, 180110, 355258, 700772, 1382287, 2726609, 5378336, 10608928, 20926496, 41278176, 81422624, 160608817, 316806289, 624911012, 1232657862, 2431458958
Offset: 1

Views

Author

Clark Kimberling, Jul 09 2011

Keywords

Comments

For discussions of polynomial reduction, see A192232 and A192744.

Examples

			The first five polynomials p(n,x) and their reductions:
F1(x)=1 -> 1
F2(x)=x -> x
F3(x)=x^2+1 -> x^2+1
F4(x)=x^3+2x -> x^2+2x+1
F5(x)=x^4+3x^2+1 -> 4x^2+1x+2, so that
A192777=(1,0,1,1,2,...), A192778=(0,1,0,2,1,...), A192779=(0,0,1,1,4,...)

Links

Crossrefs

Cf. A192744, A192232, A192616, A192780, A192781.

Programs

Formula

a(n) = a(n-1)+3*a(n-2)-a(n-3)-3*a(n-4)+a(n-5)+a(n-6).
G.f.: -x^3/(x^6+x^5-3*x^4-x^3+3*x^2+x-1). [Colin Barker, Nov 23 2012]
Showing 1-5 of 5 results.

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