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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A192387 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, 2, 4, 32, 96, 512, 1856, 8576, 33792, 147456, 602112, 2566144, 10637312, 44892160, 187269120, 787087360, 3292069888, 13812760576, 57837355008, 242497880064, 1015868817408, 4258009186304, 17841063460864, 74771320537088, 313317428035584
Offset: 1

Views

Author

Clark Kimberling, Jun 30 2011

Keywords

Comments

The polynomial p(n,x) is defined by ((x+d)^n - (x-d)^n)/(2*d), where d = sqrt(x+5). For an introduction to reductions of polynomials by substitutions such as x^2 -> x+1, see A192232.

Examples

			The first five polynomials p(n,x) and their reductions are as follows:
  p(0,x) = 1 -> 1
  p(1,x) =     2*x -> 2*x
  p(2,x) = 3 +   x +  3*x^2 -> 8 + 4*x
  p(3,x) =    12*x +  4*x^2 +  4*x^3 -> 8 + 32*x
  p(4,x) = 9 + 6*x + 31*x^2 + 10*x^3 + 5*x^4 -> 96 + 96*x.
From these, read A192386 = (1, 0, 8, 8, 96, ...) and a(n) = (0, 2, 4, 32, 96, ...).

Links

Crossrefs

Cf. A112576, A192232, A192386, A192388.

Programs

  • Magma
    R:=PowerSeriesRing(Integers(), 41);
[0] cat Coefficients(R!( 2*x^2/(1-2*x-12*x^2+8*x^3+16*x^4) )); // G. C. Greubel, Jul 10 2023
  • Mathematica
    (See A192386.)
LinearRecurrence[{2,12,-8,-16}, {0,2,4,32}, 40] (* G. C. Greubel, Jul 10 2023 *)
  • SageMath
    @CachedFunction
def a(n): # a = A192387
    if (n<5): return (0,0,2,4,32)[n]
    else: return 2*a(n-1) +12*a(n-2) -8*a(n-3) -16*a(n-4)
[a(n) for n in range(1,41)] # G. C. Greubel, Jul 10 2023

Formula

From Colin Barker, Dec 09 2012: (Start)
a(n) = 2*a(n-1) + 12*a(n-2) - 8*a(n-3) - 16*a(n-4).
G.f.: 2*x^2/(1-2*x-12*x^2+8*x^3+16*x^4). (End)
a(n) = 2^n*A112576(n). - R. J. Mathar, Mar 08 2021
From G. C. Greubel, Jul 10 2023: (Start)
T(n, k) = [x^k] ((x+sqrt(x+5))^n - (x-sqrt(x+5))^n)/(2*sqrt(x+5)).
a(n) = Sum_{k=0..n-1} T(n, k)*Fibonacci(k). (End)

A192388 a(n) = A192387(n)/2.

Original entry on oeis.org

0, 1, 2, 16, 48, 256, 928, 4288, 16896, 73728, 301056, 1283072, 5318656, 22446080, 93634560, 393543680, 1646034944, 6906380288, 28918677504, 121248940032, 507934408704, 2129004593152, 8920531730432, 37385660268544, 156658714017792
Offset: 1

Views

Author

Clark Kimberling, Jun 30 2011

Keywords

Examples

			The first five polynomials p(n,x) and their reductions are as follows:
  p(0,x) = 1 -> 1
  p(1,x) =     2*x -> 2*x
  p(2,x) = 3 +   x +  3*x^2 -> 8 + 4*x
  p(3,x) =    12*x +  4*x^2 +  4*x^3 -> 8 + 32*x
  p(4,x) = 9 + 6*x + 31*x^2 + 10*x^3 + 5*x^4 -> 96 + 96*x.
From these, read a(n) = (0, 2, 4, 32, 96, ...)/2 = (0, 1, 2, 16, 48, ...).

Links

Crossrefs

Cf. A112576, A192232, A192386, A192387.

Programs

  • Magma
    R:=PowerSeriesRing(Integers(), 41);
[0] cat Coefficients(R!( x^2/(1-2*x-12*x^2+8*x^3+16*x^4) )); // G. C. Greubel, Jul 10 2023
  • Mathematica
    (* See A192386 *)
LinearRecurrence[{2,12,-8,-16}, {0,1,2,16}, 40] (* G. C. Greubel, Jul 10 2023 *)
  • SageMath
    @CachedFunction
def a(n): # a = A192388
    if (n<5): return (0,0,1,2,16)[n]
    else: return 2*a(n-1) +12*a(n-2) -8*a(n-3) -16*a(n-4)
[a(n) for n in range(1,41)] # G. C. Greubel, Jul 10 2023

Formula

From G. C. Greubel, Jul 10 2023: (Start)
T(n, k) = [x^k] ((x+sqrt(x+5))^n - (x-sqrt(x+5))^n)/(2*sqrt(x+5)).
a(n) = (1/2)*Sum_{k=0..n-1} T(n, k)*Fibonacci(k-1).
a(n) = 2*a(n-1) + 12*a(n-2) - 8*a(n-3) - 16*a(n-4).
G.f.: x^2/(1-2*x-12*x^2+8*x^3+16*x^4).
a(n) = 2^(n-1)*A112576(n). (End)

A192423 Constant term of the reduction by x^2 -> x+2 of the polynomial p(n,x) defined below in Comments.

Original entry on oeis.org

2, 0, 4, 2, 16, 20, 78, 140, 416, 878, 2324, 5280, 13282, 31200, 76724, 182962, 445376, 1069300, 2591118, 6239980, 15089776, 36389278, 87917284, 212144640, 512334722, 1236606720, 2985883684, 7207831202, 17402424496, 42011258900
Offset: 0

Views

Author

Clark Kimberling, Jun 30 2011

Keywords

Comments

The polynomial p(n,x) is defined by ((x+d)/2)^n + ((x-d)/2)^n, where d = sqrt(x^2+4). For an introduction to reductions of polynomials by substitutions such as x^2 -> x+2, see A192232.

Examples

			The first five polynomials p(n,x) and their reductions are as follows:
  p(0,x) = 2 -> 2
  p(1,x) = x -> x
  p(2,x) = 2 + x^2 -> 4 + x
  p(3,x) = 3*x + x^3 -> 2 + 6*x
  p(4,x) = 2 + 4*x^2 + x^4 -> 16 + 9*x.
From these, read a(n) = (2, 0, 4, 2, 16, ...) and A192424 = (0, 1, 1, 6, 9, ...).

Links

Crossrefs

Cf. A000032, A000129, A078008, A161514, A192232, A192424, A192425.

Programs

  • Magma
    R:=PowerSeriesRing(Integers(), 40); Coefficients(R!( 2*(1+x)*(1-2*x)/((1+x-x^2)*(1-2*x-x^2)) )); // G. C. Greubel, Jul 11 2023
  • Mathematica
    q[x_]:= x+2; d= Sqrt[x^2+4];
p[n_, x_]:= ((x+d)/2)^n + ((x-d)/2)^n (* A161514 *)
Table[Expand[p[n, x]], {n, 0, 6}]
reductionRules = {x^y_?EvenQ -> q[x]^(y/2), x^y_?OddQ -> x q[x]^((y - 1)/2)};
t= Table[FixedPoint[Expand[#1/. reductionRules] &, p[n,x]], {n,0,30}]
Table[Coefficient[Part[t, n], x, 0], {n,30}]   (* A192423 *)
Table[Coefficient[Part[t, n], x, 1], {n,30}]   (* A192424 *)
Table[Coefficient[Part[t, n]/2, x, 1], {n,30}]   (* A192425 *)
LinearRecurrence[{1,4,-1,-1}, {2,0,4,2}, 40] (* G. C. Greubel, Jul 11 2023 *)
  • SageMath
    @CachedFunction
def a(n): # a = A192423
    if (n<4): return (2,0,4,2)[n]
    else: return a(n-1) +4*a(n-2) -a(n-3) -a(n-4)
[a(n) for n in range(41)] # G. C. Greubel, Jul 11 2023

Formula

From Colin Barker, May 11 2014: (Start)
a(n) = a(n-1) + 4*a(n-2) - a(n-3) - a(n-4).
G.f.: 2*(1+x)*(1-2*x) / ((1+x-x^2)*(1-2*x-x^2)). (End)
From G. C. Greubel, Jul 11 2023: (Start)
a(n) = Sum_{j=0..n} T(n, j)*A078008(j), where T(n, k) = [x^k] ((x + sqrt(x^2+4))^n + (x - sqrt(x^2+4))^n)/2^n.
a(n) = (2/3)*((-1)^n*A000032(n) + A000129(n+1) - A000129(n)). (End)

A192424 a(n) = A192423(n)/2.

Original entry on oeis.org

1, 0, 2, 1, 8, 10, 39, 70, 208, 439, 1162, 2640, 6641, 15600, 38362, 91481, 222688, 534650, 1295559, 3119990, 7544888, 18194639, 43958642, 106072320, 256167361, 618303360, 1492941842, 3603915601, 8701212248, 21005629450
Offset: 0

Views

Author

Clark Kimberling, Jun 30 2011

Keywords

Examples

			The first five polynomials p(n,x) and their reductions are as follows:
  p(0,x) = 2 -> 2
  p(1,x) = x -> x
  p(2,x) = 2 + x^2 -> 4 + x
  p(3,x) = 3*x + x^3 -> 2 + 6*x
  p(4,x) = 2 + 4*x^2 + x^4 -> 16 + 9*x.
From these, read A192423(n) = 2*a(n) = (2, 0, 4, 2, 16, ...) and A192425 = (0, 1, 1, 6, 9, ...).

Links

Crossrefs

Cf. A000032, A000129, A078008, A161514, A192232, A192423, A192425.

Programs

  • Magma
    R:=PowerSeriesRing(Integers(), 40); Coefficients(R!( (1+x)*(1-2*x)/((1+x-x^2)*(1-2*x-x^2)) )); // G. C. Greubel, Jul 12 2023
  • Mathematica
    (See A192423.)
LinearRecurrence[{1,4,-1,-1}, {1,0,2,1}, 40] (* G. C. Greubel, Jul 12 2023 *)
  • SageMath
    @CachedFunction
def a(n): # a = A192424
    if (n<4): return (1,0,2,1)[n]
    else: return a(n-1) +4*a(n-2) -a(n-3) -a(n-4)
[a(n) for n in range(41)] # G. C. Greubel, Jul 12 2023

Formula

From G. C. Greubel, Jul 11 2023: (Start)
a(n) = (1/2)*Sum_{j=0..n} T(n, j)*A078008(j), where T(n, k) = [x^k] ((x + sqrt(x^2+4))^n + (x - sqrt(x^2+4))^n)/2^n.
a(n) = (1/3)*((-1)^n*A000032(n) + A000129(n+1) - A000129(n)).
a(n) = a(n-1) + 4*a(n-2) - a(n-3) - a(n-4).
G.f.: (1+x)*(1-2*x)/((1+x-x^2)*(1-2*x-x^2)). (End)

A192466 Coefficient of x in the reduction by x^2->x+2 of the polynomial p(n,x)=1+x^n+x^(2n).

Original entry on oeis.org

2, 6, 24, 90, 352, 1386, 5504, 21930, 87552, 349866, 1398784, 5593770, 22372352, 89483946, 357924864, 1431677610, 5726666752, 22906579626, 91626143744, 366504225450, 1466016202752, 5864063412906, 23456250855424, 93824997829290
Offset: 1

Views

Author

Clark Kimberling, Jul 01 2011

Keywords

Comments

For an introduction to reductions of polynomials by substitutions such as x^2->x+1, see A192232.

Examples

			The first four polynomials p(n,x) and their reductions are as follows:
p(1,x)=1+x+x^2 -> 3+2x
p(2,x)=1+x^2+x^4 -> 9+6x
p(3,x)=1+x^3+x^6 -> 25+24x
p(4,x)=1+x^4+x^8 -> 93+90x.
From these, read
A192465=(3,9,25,93,...) and A192466=(2,6,24,90,...)

Links

Crossrefs

Cf. A192232, A192465, A192467.

Programs

Formula

Empirical G.f.: -2*x*(x^2 - 3*x + 1) / ((x - 1)*(x + 1)*(2*x - 1)*(4*x - 1)). - Colin Barker, Nov 12 2012
Conjectures from Colin Barker, Feb 14 2017: (Start)
a(n) = (-1 - (-1)^n + 2^n + 4^n) / 3.
a(n) = 6*a(n-1) - 7*a(n-2) - 6*a(n-3) + 8*a(n-4) for n>4.
(End)

A192468 Constant term of the reduction by x^2->x+3 of the polynomial p(n,x)=1+x^n+x^(2n).

Original entry on oeis.org

4, 16, 61, 304, 1546, 8107, 42748, 226240, 1198645, 6353944, 33688474, 178631251, 947215924, 5022815920, 26634734125, 141237718720, 748951245034, 3971518837243, 21060069709228, 111676816254688, 592197081386533, 3140288211876136
Offset: 1

Views

Author

Clark Kimberling, Jul 01 2011

Keywords

Comments

For an introduction to reductions of polynomials by substitutions such as x^2->x+3, see A192232.

Examples

			The first four polynomials p(n,x) and their reductions are as follows:
p(1,x)=1+x+x^2 -> 4+2x
p(2,x)=1+x^2+x^4 -> 16+8x
p(3,x)=1+x^3+x^6 -> 61+44x
p(4,x)=1+x^4+x^8 -> 304+224x.
From these, read
A192468=(4,16,61,304,...) and A192469=(2,8,44,224,...)

Crossrefs

Cf. A192232, A192468, A192464, A192465.

Programs

  • Mathematica
    Remove["Global`*"];
q[x_] := x + 3; p[n_, x_] := 1 + x^n + x^(2 n);
Table[Simplify[p[n, x]], {n, 1, 5}]
reductionRules = {x^y_?EvenQ -> q[x]^(y/2),
   x^y_?OddQ -> x q[x]^((y - 1)/2)};
t = Table[FixedPoint[Expand[#1 /. reductionRules] &, p[n, x]], {n, 1, 30}]
Table[Coefficient[Part[t, n], x, 0], {n, 1, 30}]
(* A192468 *)
Table[Coefficient[Part[t, n], x, 1], {n, 1, 30}]
(* A192469 *)
Table[Coefficient[Part[t, n]/2, x, 1], {n, 1, 30}]
(* A192470 *)

Formula

Empirical G.f.: -x*(81*x^4-87*x^3-x^2+20*x-4)/((x-1)*(3*x^2+x-1)*(9*x^2-7*x+1)). [Colin Barker, Nov 12 2012]

A192469 Coefficient of x in the reduction by x^2->x+3 of the polynomial p(n,x)=1+x^n+x^(2n).

Original entry on oeis.org

2, 8, 44, 224, 1178, 6200, 32786, 173600, 919988, 4877072, 25858754, 137115440, 727074530, 3855471416, 20444603516, 108412922240, 574888887530, 3048505597160, 16165538467442, 85722217226576, 454565670533252, 2410459729834544
Offset: 1

Views

Author

Clark Kimberling, Jul 01 2011

Keywords

Comments

For an introduction to reductions of polynomials by substitutions such as x^2->x+3, see A192232.

Examples

			The first four polynomials p(n,x) and their reductions are as follows:
p(1,x)=1+x+x^2 -> 4+2x
p(2,x)=1+x^2+x^4 -> 16+8x
p(3,x)=1+x^3+x^6 -> 61+44x
p(4,x)=1+x^4+x^8 -> 304+224x.
From these, read
A192468=(4,16,61,304,...) and A192469=(2,8,44,224,...)

Crossrefs

Cf. A192232, A192468.

Programs

Formula

Empirical G.f.: -2*x*(x-1)*(3*x-1)/((3*x^2+x-1)*(9*x^2-7*x+1)). [Colin Barker, Nov 12 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]

A192752 Constant term of the reduction by x^2->x+1 of the polynomial p(n,x) defined below in Comments.

Original entry on oeis.org

1, 7, 12, 23, 39, 66, 109, 179, 292, 475, 771, 1250, 2025, 3279, 5308, 8591, 13903, 22498, 36405, 58907, 95316, 154227, 249547, 403778, 653329, 1057111, 1710444, 2767559, 4478007, 7245570, 11723581, 18969155, 30692740, 49661899, 80354643
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)+4n+3 for n>0, where p(0,x)=1. For discussions of polynomial reduction, see A192232 and A192744.

Crossrefs

Cf. A192744, A192232, A192753.

Programs

  • Mathematica
    q = x^2; s = x + 1; z = 40;
p[0, n_] := 1; p[n_, x_] := x*p[n - 1, x] + 4 n + 3;
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}](* A192752 *)
u2 = Table[Coefficient[Part[t, n], x, 1], {n, 1, z}](* A192753 *)

Formula

Conjecture: G.f.: ( 1+5*x-2*x^2 ) / ( (x-1)*(x^2+x-1) ). a(n) = A000071(n+3)+5*A000071(n+2) -2*A000071(n+1) and first differences in A022136. - R. J. Mathar, May 04 2014
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