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.

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A192755 Coefficient of x in the reduction by x^2->x+1 of the polynomial p(n,x) defined below in Comments.

Original entry on oeis.org

0, 1, 7, 19, 42, 82, 150, 263, 449, 753, 1248, 2052, 3356, 5469, 8891, 14431, 23398, 37910, 61394, 99395, 160885, 260381, 421372, 681864, 1103352, 1785337, 2888815, 4674283, 7563234, 12237658, 19801038, 32038847, 51840041, 83879049
Offset: 0

Views

Author

Clark Kimberling, Jul 09 2011

Keywords

Comments

The titular polynomial is defined recursively by p(n,x)=x*(n-1,x)+5n+1 for n>0, where p(0,x)=1. For discussions of polynomial reduction, see A192232 and A192744.

Crossrefs

Cf. A192754, A192744, A192232.

Programs

  • Mathematica
    p[0, n_] := 1; p[n_, x_] := x*p[n - 1, x] + 5 n + 1;
Table[Expand[p[n, x]], {n, 0, 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, 0, z}];
u1 = Table[Coefficient[Part[t, n], x, 0], {n, 1, z}]
  (* A192754 *)
u2 = Table[Coefficient[Part[t, n], x, 1], {n, 1, z}]
  (* A192755 *)

Formula

From R. J. Mathar, May 04 2014: (Start)
Conjecture: G.f.: -x*(1+4*x) / ( (x^2+x-1)*(x-1)^2 ).
a(n) = A001924(n)+4*A001924(n-1).
Partial sums of A192754. (End)

A192773 Coefficient of x in the reduction of the n-th Fibonacci polynomial by x^3->x^2+2x+1.

Original entry on oeis.org

0, 1, 0, 4, 3, 18, 30, 98, 219, 596, 1464, 3783, 9540, 24328, 61740, 156985, 398904, 1013772, 2576475, 6547574, 16640382, 42288806, 107473443, 273129468, 694130016, 1764047839, 4483130424, 11393354512, 28954911624, 73585574049
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. A192232, A192744, A192772.

Programs

  • Mathematica
    (See A192772.)
LinearRecurrence[{1,5,-1,-5,1,1},{0,1,0,4,3,18},40] (* Harvey P. Dale, Aug 07 2025 *)

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^2*(x^2+x-1)/(x^6+x^5-5*x^4-x^3+5*x^2+x-1). [Colin Barker, Nov 23 2012]

A192774 Coefficient of x^2 in the reduction of the n-th Fibonacci polynomial by x^3->x^2+2x+1.

Original entry on oeis.org

0, 0, 1, 1, 6, 10, 34, 74, 206, 499, 1301, 3264, 8348, 21152, 53828, 136720, 347533, 883157, 2244462, 5704094, 14496130, 36840606, 93625542, 237939591, 604694601, 1536764208, 3905506648, 9925401280, 25224262440, 64104575344
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. A192232, A192744, A192772.

Programs

  • Mathematica
    (See A192772.)
LinearRecurrence[{1,5,-1,-5,1,1},{0,0,1,1,6,10},30] (* Harvey P. Dale, Jun 25 2017 *)

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^3/(x^6+x^5-5*x^4-x^3+5*x^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]

A192801 Constant term in the reduction of the polynomial (x+2)^n by x^3->x^2+x+1. See Comments.

Original entry on oeis.org

1, 2, 4, 9, 25, 84, 312, 1199, 4637, 17906, 68976, 265249, 1019069, 3913484, 15026092, 57690143, 221487945, 850350482, 3264725772, 12534190569, 48122302705, 184755243892, 709328262928, 2723314511871, 10455585321989, 40141990468066
Offset: 0

Views

Author

Clark Kimberling, Jul 10 2011

Keywords

Comments

For discussions of polynomial reduction, see A192232 and A192744.
If the same reduction is applied to the sequence (x+1)^n instead of (x+2)^n, the resulting three coefficient sequences are essentially as follows:
A078484: constants
A099216: coefficients of x
A115390: coefficients of x^2.

Examples

			The first five polynomials p(n,x) and their reductions:
p(1,x)=1 -> 1
p(2,x)=x+2 -> x+2
p(3,x)=x^2+4x+4 -> x^2+1
p(4,x)=x^3+6x^2+12x+8 -> x^2+4x+4
p(5,x)=x^4+8x^3+24x^2+32x+16 -> 7x^2+13*x+9, so that
A192798=(1,2,4,9,25,...), A192799=(0,1,4,13,42,...), A192800=(0,0,1,7,34,...).

Links

Crossrefs

Cf. A192744, A192232, A192616, A192802, A192803.

Programs

  • Mathematica
    q = x^3; s = x^2 + x + 1; z = 40;
p[n_, x_] := (x + 2)^n;
Table[Expand[p[n, x]], {n, 0, 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, 0, z}];
u1 = Table[Coefficient[Part[t, n], x, 0], {n, 1, z}]  (* A192801 *)
u2 = Table[Coefficient[Part[t, n], x, 1], {n, 1, z}]
  (* A192802 *)
u3 = Table[Coefficient[Part[t, n], x, 2], {n, 1, z}]
  (* A192803 *)

Formula

a(n) = 7*a(n-1)-15*a(n-2)+11*a(n-3).
G.f.: -(5*x^2-5*x+1)/(11*x^3-15*x^2+7*x-1). [Colin Barker, Jul 27 2012]

Extensions

Recurrence corrected by Colin Barker, Jul 27 2012

A192804 Constant term in the reduction of the polynomial 1+x+x^2+...+x^n by x^3->x^2+x+1. See Comments.

Original entry on oeis.org

1, 1, 1, 2, 3, 5, 9, 16, 29, 53, 97, 178, 327, 601, 1105, 2032, 3737, 6873, 12641, 23250, 42763, 78653, 144665, 266080, 489397, 900141, 1655617, 3045154, 5600911, 10301681, 18947745, 34850336, 64099761, 117897841, 216847937, 398845538
Offset: 0

Views

Author

Clark Kimberling, Jul 10 2011

Keywords

Comments

For discussions of polynomial reduction, see A192232 and A192744.
This sequence provides the most-significant place-values in the construction of a tribonacci code. - James Dow Allen, Jul 12 2021

Examples

			The first five polynomials p(n,x) and their reductions:
  p(1,x)=1 -> 1,
  p(2,x)=x+1 -> x+1,
  p(3,x)=x^2+x+1 -> x^2+x+1,
  p(4,x)=x^3+x^2+x+1 -> 2x^2+2x+2,
  p(5,x)=x^4+x^3+x^2+x+1 -> 4x^2+4*x+3, so that
A192804=(1,1,1,2,3,...), A000073=(0,1,1,2,4,...), A008937=(0,0,1,2,4,...).

Links

Crossrefs

Cf. A192744, A192232, A000073.

Programs

  • Mathematica
    q = x^3; s = x^2 + x + 1; z = 40;
p[0, x_] := 1; p[n_, x_] := x^n + p[n - 1, x];
Table[Expand[p[n, x]], {n, 0, 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, 0, z}];
u1 = Table[Coefficient[Part[t, n], x, 0], {n, 1, z}]
  (* A192804 *)
u2 = Table[Coefficient[Part[t, n], x, 1], {n, 1, z}]
  (* A000073 *)
u3 = Table[Coefficient[Part[t, n], x, 2], {n, 1, z}]
  (* A008937 *)

Formula

a(n) = 2*a(n-1) - a(n-4).
a(n) = a(n-1) + a(n-2) + a(n-3) - 1. - Alzhekeyev Ascar M, Feb 05 2012
G.f.: ( 1-x-x^2 ) / ( (x-1)*(x^3+x^2+x-1) ). - R. J. Mathar, May 06 2014
a(n) - a(n-1) = A000073(n-1). - R. J. Mathar, May 06 2014

A192809 Coefficient of x in the reduction of the polynomial (x^2 + 2)^n by x^3 -> x^2 + 2.

Original entry on oeis.org

0, 0, 2, 14, 74, 366, 1786, 8702, 42410, 206734, 1007834, 4913310, 23953034, 116774190, 569289402, 2775359806, 13530239338, 65961672910, 321571716762, 1567703857118, 7642759781962, 37259445922414, 181644634930298, 885541171698814
Offset: 0

Views

Author

Clark Kimberling, Jul 10 2011

Keywords

Comments

For discussions of polynomial reduction, see A192232 and A192744.

Links

Crossrefs

Cf. A192744, A192232, A192808, A192811.

Programs

  • GAP
    a:=[0,0,2];; for n in [4..25] do a[n]:=7*a[n-1]-12*a[n-2]+8*a[n-3]; od; Print(a); # Muniru A Asiru, Jan 02 2019
  • Magma
    m:=30; R:=PowerSeriesRing(Integers(), m); [0,0] cat Coefficients(R!( 2*x^2/(1-7*x+12*x^2-8*x^3) )); // G. C. Greubel, Jan 02 2019
  • Mathematica
    (See A192808.)
LinearRecurrence[{7,-12,8}, {0,0,2}, 30] (* G. C. Greubel, Jan 02 2019 *)
  • PARI
    my(x='x+O('x^30)); concat([0,0], Vec(2*x^2/(1-7*x+12*x^2-8*x^3))) \\ G. C. Greubel, Jan 02 2019
  • Sage
    (2*x^2/(1-7*x+12*x^2-8*x^3)).series(x, 30).coefficients(x, sparse=False) # G. C. Greubel, Jan 02 2019

Formula

a(n) = 7*a(n-1) - 12*a(n-2) + 8*a(n-3).
a(n) = 2*A192811(n).
G.f.: 2*x^2/(1-7*x+12*x^2-8*x^3). - Colin Barker, Jul 26 2012

A192811 a(n) = A192809(n)/2.

Original entry on oeis.org

0, 0, 1, 7, 37, 183, 893, 4351, 21205, 103367, 503917, 2456655, 11976517, 58387095, 284644701, 1387679903, 6765119669, 32980836455, 160785858381, 783851928559, 3821379890981, 18629722961207, 90822317465149, 442770585849407
Offset: 0

Views

Author

Clark Kimberling, Jul 10 2011

Keywords

Links

Crossrefs

Cf. A192232, A192744, A192809.

Programs

  • GAP
    a:=[0,0,1];; for n in [4..25] do a[n]:=7*a[n-1]-12*a[n-2]+8*a[n-3]; od; Print(a); # Muniru A Asiru, Jan 03 2019
  • Magma
    m:=30; R:=PowerSeriesRing(Integers(), m); [0,0] cat Coefficients(R!( x^2/(1-7*x+12*x^2-8*x^3) )); // G. C. Greubel, Jan 03 2019
  • Mathematica
    (See A192808.)
LinearRecurrence[{7,-12,8},{0,0,1},30] (* Harvey P. Dale, Dec 06 2018 *)
  • PARI
    my(x='x+O('x^30)); concat([0,0], Vec(x^2/(1-7*x+12*x^2-8*x^3))) \\ G. C. Greubel, Jan 03 2019
  • Sage
    (x^2/(1-7*x+12*x^2-8*x^3)).series(x, 30).coefficients(x, sparse=False) # G. C. Greubel, Jan 03 2019

Formula

a(n) = 7*a(n-1) - 12*a(n-2) + 8*a(n-3).
G.f.: x^2/(1-7*x+12*x^2-8*x^3). - Colin Barker, Jul 26 2012

Extensions

Name corrected by Colin Barker, Jul 26 2012

A192814 Constant term in the reduction of the polynomial (2*x+1)^n by x^3 -> x^2 + x + 1. See Comments.

Original entry on oeis.org

1, 1, 1, 9, 49, 225, 1041, 4873, 22817, 106753, 499425, 2336585, 10931921, 51145825, 239289457, 1119533257, 5237818689, 24505519873, 114650876097, 536402551689, 2509598769265, 11741342323937, 54932733173713, 257006830281609
Offset: 0

Views

Author

Clark Kimberling, Jul 10 2011

Keywords

Comments

For discussions of polynomial reduction, see A192232 and A192744.

Links

Crossrefs

Cf. A192744, A192232, A192815.

Programs

  • GAP
    a:=[1,1,1];; for n in [4..25] do a[n]:=5*a[n-1]-3*a[n-2]+7*a[n-3]; od; Print(a); # Muniru A Asiru, Jan 03 2019
  • Magma
    m:=30; R:=PowerSeriesRing(Integers(), m); Coefficients(R!( (1-4*x-x^2)/(1-5*x+3*x^2-7*x^3) )); // G. C. Greubel, Jan 03 2019
  • Maple
    seq(coeff(series((1-4*x-x^2)/(1-5*x+3*x^2-7*x^3),x,n+1), x, n), n = 0 .. 25); # Muniru A Asiru, Jan 03 2019
  • Mathematica
    q = x^3; s = x^2 + x + 1; z = 40;
p[n_, x_] := (2 x + 1)^n;
Table[Expand[p[n, x]], {n, 0, 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, 0, z}];
u1 = Table[Coefficient[Part[t, n], x, 0], {n, 1, z}] (* A192814 *)
u2 = Table[Coefficient[Part[t, n], x, 1], {n, 1, z}] (* A192815 *)
u2 = u2/2  (* A192816 *)
LinearRecurrence[{5,-3,7}, {1,1,1}, 30] (* G. C. Greubel, Jan 03 2019 *)
  • PARI
    my(x='x+O('x^30)); Vec((1-4*x-x^2)/(1-5*x+3*x^2-7*x^3)) \\ G. C. Greubel, Jan 03 2019
  • Sage
    ((1-4*x-x^2)/(1-5*x+3*x^2-7*x^3)).series(x, 30).coefficients(x, sparse=False) # G. C. Greubel, Jan 03 2019

Formula

a(n) = 5*a(n-1) - 3*a(n-2) + 7*a(n-3).
G.f.: (1 -4*x -x^2) / (1 -5*x +3*x^2 -7*x^3). - R. J. Mathar, May 06 2014
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