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 1-10 of 13 results. Next

A192772 Constant term in the reduction of the n-th Fibonacci polynomial by x^3->x^2+2x+1.

Original entry on oeis.org

1, 0, 1, 1, 2, 7, 12, 41, 86, 247, 585, 1548, 3849, 9896, 25001, 63724, 161721, 411257, 1044878, 2655719, 6748972, 17151849, 43589578, 110777391, 281529169, 715471992, 1818293377, 4620978640, 11743694657, 29845241080, 75848270001
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+4x+1
F5(x)=x^4+3x^2+1 -> 6x^2+3x+2, so that
A192772=(1,0,1,1,2,...), A192773=(0,1,0,4,3,...), A192774=(0,0,1,1,6,...)

Links

Crossrefs

Cf. A192744, A192232, A192616, A192773, A192774.

Programs

  • Mathematica
    q = x^3; s = x^2 + 2 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}] (* A192772 *)
u2 = Table[Coefficient[Part[t, n], x, 1], {n, 1, z}] (* A192773 *)
u3 = Table[Coefficient[Part[t, n], x, 2], {n, 1, z}] (* A192774 *)

Formula

a(n) = a(n-1)+5*a(n-2)-a(n-3)-5*a(n-4)+a(n-5)+a(n-6).
G.f.: -x*(x^2-x-1)*(x^2+2*x-1) / (x^6+x^5-5*x^4-x^3+5*x^2+x-1). [Colin Barker, Jan 17 2013]

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]

A192779 Coefficient of x^2 in the reduction of the n-th Fibonacci polynomial by x^3->x^2+3x+1.

Original entry on oeis.org

0, 0, 1, 1, 7, 12, 47, 107, 337, 868, 2520, 6808, 19192, 52756, 147185, 407069, 1131599, 3136292, 8707655, 24151335, 67025633, 185946904, 515971328, 1431563056, 3972149312, 11021051864, 30579529249, 84846231017, 235416993159, 653192251196
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.

Programs

  • Mathematica
    (See A192777.)
LinearRecurrence[{1,6,-1,-6,1,1},{0,0,1,1,7,12}, 30] (* Harvey P. Dale, Oct 29 2018 *)

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^3/((x^2-x-1)*(x^4+2*x^3-3*x^2-2*x+1)). [Colin Barker, Nov 23 2012]

A192798 Constant term in the reduction of the n-th Fibonacci polynomial by x^3->x^2+2. See Comments.

Original entry on oeis.org

1, 0, 1, 2, 3, 10, 17, 42, 87, 188, 411, 876, 1907, 4100, 8863, 19134, 41289, 89174, 192459, 415542, 897049, 1936576, 4180809, 9025544, 19484825, 42064320, 90809993, 196043706, 423225563, 913674090, 1972469945, 4258235410, 9192822255
Offset: 1

Views

Author

Clark Kimberling, Jul 10 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+2
F5(x)=x^4+3x^2+1 -> 4x^2+2*x+3, so that
A192798=(1,0,1,2,3,...), A192799=(0,1,0,2,2,...), A192800=(0,0,1,1,4,...)

Links

Crossrefs

Cf. A192744, A192232, A192616, A192799, A192800.

Programs

  • Mathematica
    q = x^3; s = x^2 + 2; 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}]  (* A192798 *)
u2 = Table[Coefficient[Part[t, n], x, 1], {n, 1, z}]
  (* A192799 *)
u3 = Table[Coefficient[Part[t, n], x, 2], {n, 1, z}]
  (* A192800 *)

Formula

a(n) = a(n-1)+3*a(n-2)-3*a(n-4)+a(n-5)+a(n-6).
G.f.: -x*(x-1)*(x+1)*(x^2+x-1)/(x^6+x^5-3*x^4+3*x^2+x-1). [Colin Barker, Jul 27 2012]

Extensions

Comment in Mathematica code corrected by Colin Barker, Jul 27 2012

A192800 Coefficient of x^2 in the reduction of the n-th Fibonacci polynomial by x^3->x^2+2.

Original entry on oeis.org

0, 0, 1, 1, 4, 7, 16, 35, 73, 162, 344, 748, 1612, 3478, 7517, 16213, 35020, 75585, 163184, 352295, 760517, 1641880, 3544484, 7652008, 16519388, 35662584, 76989693, 166207785, 358815192, 774622191, 1672280660, 3610176155, 7793770037
Offset: 1

Views

Author

Clark Kimberling, Jul 10 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+2
F5(x)=x^4+3x^2+1 -> 4x^2+2*x+3, so that
A192798=(1,0,1,2,3,...), A192799=(0,1,0,2,2,...), A192800=(0,0,1,1,4,...)

Links

Crossrefs

Cf. A192744, A192232, A192616, A192798, A192800.

Programs

Formula

a(n) = a(n-1)+3*a(n-2)-3*a(n-4)+a(n-5)+a(n-6).
G.f.: -x^3/(x^6+x^5-3*x^4+3*x^2+x-1). [Colin Barker, Jul 27 2012]

A192617 Coefficient of x in the reduction of the n-th Fibonacci polynomial by x^3->x^2+x+1.

Original entry on oeis.org

0, 1, 0, 3, 2, 10, 16, 43, 92, 213, 486, 1100, 2522, 5719, 13068, 29721, 67772, 154334, 351670, 801137, 1825184, 4158219, 9473244, 21582392, 49169220, 112018989, 255203904, 581412535, 1324587918, 3017709810, 6875021540, 15662845615
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+3x+1
F5(x)=x^4+3x^2+1 -> 4x^2+2x+2, so that
A192616=(1,0,1,1,2,...), A192617=(0,1,0,3,2,...), A192651=(0,0,1,1,5,...)

Links

Crossrefs

Cf. A192744, A192232, A192616.

Programs

  • Mathematica
    (See A192616.)
LinearRecurrence[{1,4,-1,-4,1,1},{0,1,0,3,2,10},40] (* Harvey P. Dale, Feb 23 2021 *)

Formula

a(n) = a(n-1)+4*a(n-2)-a(n-3)-4a(n-4)+a(n-5)+a(n-6).
G.f.: x^2*(x^2+x-1)/(x^6+x^5-4*x^4-x^3+4*x^2+x-1). [Colin Barker, Jul 27 2012]

A192651 Coefficient of x^2 in the reduction of the n-th Fibonacci polynomial by x^3->x^2+x+1. See Comments.

Original entry on oeis.org

0, 0, 1, 1, 5, 8, 23, 47, 113, 252, 578, 1316, 2994, 6832, 15545, 35445, 80711, 183928, 418973, 954571, 2174681, 4954436, 11287336, 25715016, 58584744, 133468980, 304072713, 692745597, 1578230845, 3595564360, 8191505015, 18662090915
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+3x+1
F5(x)=x^4+3x^2+1 -> 4x^2+2x+2, so that
A192616=(1,0,1,1,2,...), A192617=(0,1,0,3,2,...), A192651=(0,0,1,1,5,...)

Links

Crossrefs

Cf. A192232, A192744, A192616.

Programs

Formula

a(n) = a(n-1)+4*a(n-2)-a(n-3)-4a(n-4)+a(n-5)+a(n-6).
G.f.: -x^3/(x^6+x^5-4*x^4-x^3+4*x^2+x-1). [Colin Barker, Jul 27 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]
Showing 1-10 of 13 results. Next

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