A161516 -id:A161516 - cp's OEIS frontend

A192344 Constant term of the reduction of n-th polynomial at A161516 by x^2->x+1.

1, 0, 5, 4, 49, 108, 637, 2024, 9329, 34104, 143621, 554092, 2255809, 8883876, 35708701, 141734480, 566950433, 2257038576, 9011796293, 35916665428, 143306508433, 571395546204, 2279250017533, 9089366457656, 36253101237521, 144581807030568

Offset: 1

Clark Kimberling, Jun 28 2011

nonn

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

			The first four polynomials at A161516 and their reductions are as follows:
p0(x)=1 -> 1
p1(x)=x -> x
p2(x)=4+x+x^2 -> 5+2x
p3(x)=12x+3x^2+x^3 -> 4+17x.
From these, we read
A192344=(1,0,5,4,...) and A192345=(1,1,2,17...)

Cf. A192232, A192345.

q[x_] := x + 1; d = Sqrt[x + 4];
p[n_, x_] := ((x + d)^n + (x - d)^n )/
  2 (* polynomials defined at A161516 *)
Table[Expand[p[n, x]], {n, 0, 4}]
reductionRules = {x^y_?EvenQ -> q[x]^(y/2),
   x^y_?OddQ -> x q[x]^((y - 1)/2)};
t = Table[Last[Most[FixedPointList[Expand[#1 /. reductionRules] &, p[n, x]]]], {n, 0, 30}]
Table[Coefficient[Part[t, n], x, 0], {n, 1, 30}]
(* A192344 *)
Table[Coefficient[Part[t, n], x, 1], {n, 1, 30}]
(* A192345 *)

Conjecture a(n) = 2*a(n-1)+10*a(n-2)-6*a(n-3)-9*a(n-4). G.f.: -(5*x^2+2*x-1) / (9*x^4+6*x^3-10*x^2-2*x+1). [Colin Barker, Nov 22 2012]

A192345 Constant term of the reduction of n-th polynomial at A161516 by x^2->x+1.

Original entry on oeis.org

0, 1, 2, 17, 48, 245, 850, 3709, 14016, 57817, 225890, 912473, 3610800, 14470637, 57541426, 229912165, 915917568, 3655472497, 14572774850, 58135559777, 231822774960, 924665246117, 3687589909522, 14707675592461, 58656853828800, 233942936910025

Offset: 1

Clark Kimberling, Jun 28 2011

nonn

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

			(See A192344.)

Cf. A192232, A192344.

Mathematica
```
(See A192344.)
```

Empirical G.f.: x^2*(3*x^2+1)/(9*x^4+6*x^3-10*x^2-2*x+1). [Colin Barker, Nov 13 2012]

A192352 Constant term of the reduction of the polynomial p(n,x)=(1/2)((x+1)^n+(x-1)^n) by x^2->x+1.

Original entry on oeis.org

1, 0, 2, 1, 9, 13, 51, 106, 322, 771, 2135, 5401, 14445, 37324, 98514, 256621, 673933, 1760997, 4615823, 12075526, 31628466, 82781215, 216761547, 567428401, 1485645049, 3889310328, 10182603746, 26657986681, 69792188337, 182717232061

Offset: 1

Clark Kimberling, Jun 29 2011

nonn

1

			For an introduction to reductions of polynomials by substitutions such as x^2->x+1, see A192232.
The first six polynomials p(n,x) and their reductions are as follows:
p(0,x)=1 -> 1
p(1,x)=x -> x
p(2,x)=1+x^2 -> 2+x
p(3,x)=3x+x^3 -> 1+5x
p(4,x)=1+6x^2+x^4 -> 9+9x
p(5,x)=5x+10x^3+x^5 -> 13+30x.
From these, we read
A192352=(1,0,2,1,9,13...) and A049602=(0,1,1,5,9,30...).

Cf. A192232, A049602.

q[x_] := x + 1; d = 1;
p[n_, x_] := ((x + d)^n + (x - d)^n )/
  2 (* similar to polynomials defined at A161516 *)
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[Last[Most[FixedPointList[Expand[#1 /. reductionRules] &, p[n, x]]]], {n, 0, 30}]
Table[Coefficient[Part[t, n], x, 0], {n, 1, 30}]
  (* A192352 *)
Table[Coefficient[Part[t, n], x, 1], {n, 1, 30}]
  (* A049602 *)

Empirical G.f.: -x*(x^3-x^2-2*x+1)/((x^2-3*x+1)*(x^2-x-1)). [Colin Barker, Sep 11 2012]

A192346 Constant term of the reduction (by x^2->x+1) of polynomial p(n,x) identified in Comments.

Original entry on oeis.org

1, 0, 3, 4, 25, 68, 275, 904, 3297, 11400, 40499, 141900, 500697, 1760396, 6200675, 21820432, 76823425, 270407696, 951914403, 3350807700, 11795463001, 41521535700, 146162319603, 514512119704, 1811159622625, 6375545788568, 22442862753875

Offset: 1

Clark Kimberling, Jun 28 2011

nonn

To define the polynomials p(n,x), let d=sqrt(x+2); then p(n,x)=(1/2)((x+d)^n+(x-d)^n). These are similar to polynomials at A161516.

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

			The first four polynomials p(n,x) and their reductions are as follows:
p(0,x)=1 -> 1
p(1,x)=x -> x
p(2,x)=2+x+x^2 -> 3+2x
p(3,x)=6x+3x^2+x^3 -> 4+11x.
From these, we read
A192346=(1,0,3,4,...) and A192347=(1,1,2,11...)

Cf. A192232, A192347.

q[x_] := x + 1; d = Sqrt[x + 2];
p[n_, x_] := ((x + d)^n + (x - d)^n )/ 2
(* similar to polynomials defined at A161516 *)
Table[Expand[p[n, x]], {n, 0, 4}]
reductionRules = {x^y_?EvenQ -> q[x]^(y/2),
   x^y_?OddQ -> x q[x]^((y - 1)/2)};
t = Table[Last[Most[FixedPointList[Expand[#1 /. reductionRules] &, p[n, x]]]], {n, 0, 30}]
Table[Coefficient[Part[t, n], x, 0], {n, 1, 30}]
(* A192346 *)
Table[Coefficient[Part[t, n], x, 1], {n, 1, 30}]
(* A192347 *)

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

A192347 Coefficient of x in the reduction (by x^2->x+1) of polynomial p(n,x) identified in Comments.

Original entry on oeis.org

0, 1, 2, 11, 32, 125, 418, 1511, 5248, 18601, 65250, 230099, 809248, 2849989, 10030018, 35311375, 124293632, 437545489, 1540200002, 5421774299, 19085364000, 67183428301, 236495292002, 832498651511, 2930516834432, 10315851565625

Offset: 1

Clark Kimberling, Jun 28 2011

nonn

To define the polynomials p(n,x), let d=sqrt(x+2); then p(n,x)=(1/2)((x+d)^n+(x-d)^n). These are similar to polynomials at A161516.

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

			The first four polynomials p(n,x) and their reductions are as follows:
p(0,x)=1 -> 1
p(1,x)=x -> x
p(2,x)=2+x+x^2 -> 3+2x
p(3,x)=6x+3x^2+x^3 -> 4+11x.
From these, we read
A192346=(1,0,3,4,...) and A192347=(1,1,2,11...)

Cf. A192232, A192346.

Mathematica
```
(See A192346.)
```

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

A192348 Constant term of the reduction (by x^2->x+1) of polynomial p(n,x) identified in Comments.

Original entry on oeis.org

1, 0, 4, 4, 36, 88, 432, 1408, 5776, 20736, 80320, 297792, 1132096, 4242304, 16028928, 60276736, 227287296, 855703552, 3224482816, 12144337920, 45752574976, 172339107840, 649223532544, 2445572276224, 9212566081536, 34703459811328

Offset: 1

Clark Kimberling, Jun 28 2011

nonn

To define the polynomials p(n,x), let d=sqrt(x+3); then p(n,x)=(1/2)((x+d)^n+(x-d)^n). These are similar to polynomials at A161516.

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

			The first four polynomials p(n,x) and their reductions are as follows:
p(0,x)=1 -> 1
p(1,x)=x -> x
p(2,x)=3+x+x^2 -> 4+2x
p(3,x)=9x+3x^2+x^3 -> 4+14x.
From these, we read
A192348=(1,0,3,4,...) and A192349=(0,1,2,14...)

Cf. A192232, A192349.

q[x_] := x + 1; d = Sqrt[x + 3];
p[n_, x_] := ((x + d)^n + (x - d)^n )/
  2 (* similar to polynomials defined at A161516 *)
Table[Expand[p[n, x]], {n, 0, 4}]
reductionRules = {x^y_?EvenQ -> q[x]^(y/2),
   x^y_?OddQ -> x q[x]^((y - 1)/2)};
t = Table[Last[Most[FixedPointList[Expand[#1 /. reductionRules] &, p[n, x]]]], {n, 0, 30}]
Table[Coefficient[Part[t, n], x, 0], {n, 1, 30}]
(* A192348 *)
Table[Coefficient[Part[t, n], x, 1], {n, 1, 30}]
(* A192349 *)

Conjecture: a(n) = 2*a(n-1)+8*a(n-2)-4*a(n-3)-4*a(n-4). G.f.: -x*(4*x^2+2*x-1) / (4*x^4+4*x^3-8*x^2-2*x+1). [Colin Barker, Jan 17 2013]

A192349 Coefficient of x in the reduction (by x^2->x+1) of polynomial p(n,x) identified in Comments.

Original entry on oeis.org

0, 1, 2, 14, 40, 180, 616, 2456, 8960, 34384, 128160, 485728, 1823360, 6882368, 25896064, 97614720, 367575040, 1384954112, 5216465408, 19651804672, 74025216000, 278859191296, 1050447030272, 3957059508224, 14906157629440, 56151566438400

Offset: 1

Clark Kimberling, Jun 28 2011

nonn

To define the polynomials p(n,x), let d=sqrt(x+3); then p(n,x)=(1/2)((x+d)^n+(x-d)^n). These are similar to polynomials at A161516. For an introduction to reductions of polynomials by substitutions such as x^2->x+1, see A192232.

			The first four polynomials p(n,x) and their reductions are as follows:
p(0,x)=1 -> 1
p(1,x)=x -> x
p(2,x)=3+x+x^2 -> 4+2x
p(3,x)=9x+3x^2+x^3 -> 4+14x.
From these, we read
A192348=(1,0,3,4,...) and A192349=(0,1,2,14...)

Cf. A192232, A192348.

Mathematica
```
(See A192348.)
```

Conjecture: a(n) = 2*a(n-1)+8*a(n-2)-4*a(n-3)-4*a(n-4). G.f.: x^2*(2*x^2+1) / (4*x^4+4*x^3-8*x^2-2*x+1). [Colin Barker, Jan 17 2013]

A192350 Constant term of the reduction (by x^2->x+1) of polynomial p(n,x) identified in Comments.

Original entry on oeis.org

1, 0, 6, 4, 64, 128, 896, 2752, 14208, 52224, 238592, 946176, 4110336, 16830464, 71598080, 297140224, 1253048320, 5229707264, 21973303296, 91924463616, 385642135552, 1614916091904, 6770569248768, 28364203098112, 118885634277376

Offset: 1

Clark Kimberling, Jun 28 2011

nonn

To define the polynomials p(n,x), let d=sqrt(x+5); then p(n,x)=(1/2)((x+d)^n+(x-d)^n). These are similar to polynomials at A161516.

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

			The first four polynomials p(n,x) and their reductions are as follows:
p(0,x)=1 -> 1
p(1,x)=x -> x
p(2,x)=5+x+x^2 -> 6+2x
p(3,x)=15x+3x^2+x^3 -> 4+20x.
From these, we read
A192350=(1,0,6,4,...) and A192351=(0,1,2,20...)

Cf. A192232, A192351.

q[x_] := x + 1; d = Sqrt[x + 5];
p[n_, x_] := ((x + d)^n + (x - d)^n )/
  2 (* similar to polynomials defined at A161516 *)
Table[Expand[p[n, x]], {n, 0, 4}]
reductionRules = {x^y_?EvenQ -> q[x]^(y/2),
   x^y_?OddQ -> x q[x]^((y - 1)/2)};
t = Table[Last[Most[FixedPointList[Expand[#1 /. reductionRules] &, p[n, x]]]], {n, 0, 30}]
Table[Coefficient[Part[t, n], x, 0], {n, 1, 30}]
  (* A192350 *)
Table[Coefficient[Part[t, n], x, 1], {n, 1, 30}]
  (* A192351 *)

Conjecture: a(n) = 2*a(n-1)+12*a(n-2)-8*a(n-3)-16*a(n-4). G.f.: -x*(6*x^2+2*x-1) / (16*x^4+8*x^3-12*x^2-2*x+1). [Colin Barker, Jan 17 2013]

A192351 Coefficient of x in the reduction (by x^2->x+1) of polynomial p(n,x) identified in Comments.

Original entry on oeis.org

0, 1, 2, 20, 56, 320, 1120, 5312, 20608, 90880, 368640, 1577984, 6522880, 27578368, 114909184, 483328000, 2020573184, 8480555008, 35502817280, 148874461184, 623609118720, 2614000353280, 10952269365248, 45901678641152, 192340840939520

Offset: 0

Clark Kimberling, Jun 28 2011

nonn

To define the polynomials p(n,x), let d=sqrt(x+5); then p(n,x)=(1/2)((x+d)^n+(x-d)^n). These are similar to polynomials at A161516.

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

			The first four polynomials p(n,x) and their reductions are as follows:
p(0,x)=1 -> 1
p(1,x)=x -> x
p(2,x)=5+x+x^2 -> 6+2x
p(3,x)=15x+3x^2+x^3 -> 4+20x.
From these, we read
A192350=(1,0,6,4,...) and A192351=(0,1,2,20...)

Robert Israel, Table of n, a(n) for n = 0..1606

Cf. A192232, A192350.

f:= gfun:-rectoproc({a(n) = 2*a(n-1)+12*a(n-2)-8*a(n-3)-16*a(n-4),a(0)=0,a(1)=1,a(2)=2,a(3)=20},a(n),remember):
map(f, [$0..50]); # Robert Israel, Jan 01 2018

Mathematica
```
(See A192350.)
```

Conjecture: a(n) = 2*a(n-1)+12*a(n-2)-8*a(n-3)-16*a(n-4). G.f.: x*(4*x^2+1) / (16*x^4+8*x^3-12*x^2-2*x+1). [Colin Barker, Jan 17 2013]
Confirmation of conjecture by Robert Israel, Jan 01 2018: (Start)
The polynomials p(n,x) have g.f. G(z) = (1-x*z)/(1-2*x*z-5*z^2-x*z^2+x^2*z^2).
The reductions mod x^2-x-1 have g.f. g(z) = (1+x*z-2*z-6*z^2+4*x*z^3)/(1-2*z-12*z^2+8*z^3+16*z^4):
note that the numerator of G(z)-g(z) is divisible by x^2-x-1. (End)

Offset corrected by Robert Israel, Jan 01 2018

A192353 Constant term of the reduction of the polynomial p(n,x)=(1/2)((x+2)^n+(x-2)^n) by x^2->x+1.

Original entry on oeis.org

1, 0, 5, 1, 42, 43, 429, 820, 4861, 12597, 58598, 177859, 732825, 2417416, 9358677, 32256553, 120902914, 426440955, 1571649221, 5610955132, 20497829133, 73645557469, 267803779710, 965384509651, 3502058316337, 12646311635088, 45818284122149

Offset: 1

Clark Kimberling, Jun 29 2011

nonn

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

			(See A192352 for a related example.)

Cf. A192232, A192354, A192352.

q[x_] := x + 1; d = 2;
p[n_, x_] := ((x + d)^n + (x - d)^n )/2 (* similar to polynomials defined at A161516 *)
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[Last[Most[FixedPointList[Expand[#1 /. reductionRules] &, p[n, x]]]], {n, 0, 30}]
Table[Coefficient[Part[t,n],x,0], {n,1,30}](* A192353 *)
Table[Coefficient[Part[t,n],x,1], {n,1,30}]  (* A192354 *)

Empirical G.f.: x*(x^3-4*x^2-2*x+1)/((x^2+3*x+1)*(5*x^2-5*x+1)). [Colin Barker, Sep 11 2012]

cp's OEIS Frontend

A192344 Constant term of the reduction of n-th polynomial at A161516 by x^2->x+1.

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A192345 Constant term of the reduction of n-th polynomial at A161516 by x^2->x+1.

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A192352 Constant term of the reduction of the polynomial p(n,x)=(1/2)((x+1)^n+(x-1)^n) by x^2->x+1.

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A192346 Constant term of the reduction (by x^2->x+1) of polynomial p(n,x) identified in Comments.

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A192347 Coefficient of x in the reduction (by x^2->x+1) of polynomial p(n,x) identified in Comments.

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A192348 Constant term of the reduction (by x^2->x+1) of polynomial p(n,x) identified in Comments.

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A192349 Coefficient of x in the reduction (by x^2->x+1) of polynomial p(n,x) identified in Comments.

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Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica

Formula

A192350 Constant term of the reduction (by x^2->x+1) of polynomial p(n,x) identified in Comments.

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Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica