A162517 -id:A162517 - cp's OEIS frontend

A192373 Constant term in the reduction of the polynomial p(n,x) defined at A162517 and below in Comments.

1, 0, 7, 8, 77, 192, 1043, 3472, 15529, 57792, 240655, 934808, 3789653, 14963328, 60048443, 238578976, 953755537, 3798340224, 15162325975, 60438310184, 241126038941, 961476161856, 3835121918243, 15294304429744, 61000836720313, 243280700771904

Offset: 1

Clark Kimberling, Jun 29 2011

nonn

The polynomials are given by p(n,x)=((x+d)^n-(x-d)^n)/(2d), where d=sqrt(x+4).

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

			The first five polynomials p(n,x) and their reductions are as follows:
p(0,x)=1 -> 1
p(1,x)=2x -> 2x
p(2,x)=4+x+3x^2 -> 7+4x
p(3,x)=16x+4x^2+4x^3 -> 8+28x
p(4,x)=16+8x+41x^2+10x^3+5x^4 -> 77+84x.
From these, read A192352=(1,0,7,8,77,...) and A049602=(0,2,4,28,84,...).

Cf. A192232, A192374, A192375, A162517.

q[x_] := x + 1; d = Sqrt[x + 4];
p[n_, x_] := ((x + d)^n - (x - d)^n )/(2 d) (* A162517 *)
Table[Expand[p[n, x]], {n, 1, 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, 1, 30}]
Table[Coefficient[Part[t, n], x, 0], {n, 1, 30}]  (* A192373 *)
Table[Coefficient[Part[t, n], x, 1], {n, 1, 30}]  (* A192374 *)
Table[Coefficient[Part[t, n]/2, x, 1], {n, 1, 30}] (* A192375 *)

Conjecture: a(n) = 2*a(n-1)+10*a(n-2)-6*a(n-3)-9*a(n-4). G.f.: -x*(x+1)*(3*x-1) / (9*x^4+6*x^3-10*x^2-2*x+1). - Colin Barker, May 09 2014

A192374 Coefficient of x in the reduction of the polynomial p(n,x) defined at A162517 and below in Comments.

Original entry on oeis.org

0, 2, 4, 28, 84, 406, 1448, 6200, 23688, 97034, 380716, 1533844, 6079452, 24339742, 96844496, 386805104, 1541301648, 6150529682, 24521644756, 97819530508, 390080615652, 1555871900710, 6204937972088, 24747735482792, 98698893741336

Offset: 1

Clark Kimberling, Jun 29 2011

nonn

The polynomials p(n,x) is defined by ((x+d)^n-(x-d)^n)/(2d), where d=sqrt(x+4). A192374=2*A192375. For an introduction to reductions of polynomials by substitutions such as x^2->x+1, see A192232.

			(See A192373.)

Cf. A192232, A192373, A192375.

Mathematica
```
(See A192373.)
```

Empirical G.f.: 2*x^2/(9*x^4+6*x^3-10*x^2-2*x+1). [Colin Barker, Nov 23 2012]

A162515 Triangle of coefficients of polynomials defined by Binet form: P(n,x) = (U^n - L^n)/d, where U = (x + d)/2, L = (x - d)/2, d = sqrt(x^2 + 4).

Original entry on oeis.org

0, 1, 1, 0, 1, 0, 1, 1, 0, 2, 0, 1, 0, 3, 0, 1, 1, 0, 4, 0, 3, 0, 1, 0, 5, 0, 6, 0, 1, 1, 0, 6, 0, 10, 0, 4, 0, 1, 0, 7, 0, 15, 0, 10, 0, 1, 1, 0, 8, 0, 21, 0, 20, 0, 5, 0, 1, 0, 9, 0, 28, 0, 35, 0, 15, 0, 1, 1, 0, 10, 0, 36, 0, 56, 0, 35, 0, 6, 0, 1, 0, 11, 0, 45, 0, 84, 0, 70, 0, 21, 0, 1

Offset: 0

Clark Kimberling, Jul 05 2009

Row sums 0,1,1,2,3,5,... are the Fibonacci numbers, A000045.
Note that the coefficients are given in decreasing order. - M. F. Hasler, Dec 07 2011
Essentially a mirror image of A168561. - Philippe Deléham, Dec 08 2013

			Polynomial expansion:
  0;
  1;
  x;
  x^2 + 1;
  x^3 + 2*x;
  x^4 + 3*x^2 + 1;
First rows:
  0;
  1;
  1, 0;
  1, 0, 1;
  1, 0, 2, 0;
  1, 0, 3, 0, 1;
  1, 0, 4, 0, 3, 0;
Row 6 matches P(6,x)=x^5 + 4*x^3 + 3*x.

G. C. Greubel, Rows n = 0..101 of triangle
T. Copeland, Addendum to Elliptic Lie Triad

Cf. A000045, A049310, A053119, A162514, A162516, A162517.

T:= function(n,k)
    if (k mod 2)=0 then return Binomial(n- k/2, k/2);
    else return 0;
    fi; end;
Concatenation([0], Flat(List([0..15], n-> List([0..n], k-> T(n,k) ))) ); # G. C. Greubel, Jan 01 2020

function T(n,k)
  if (k mod 2) eq 0 then return Round( Gamma(n-k/2+1)/(Gamma(k/2+1)*Gamma(n-k+1)));
  else return 0;
  end if; return T; end function;
[0] cat [T(n,k): k in [0..n], n in [0..15]]; // G. C. Greubel, Jan 01 2020

0, seq(seq(`if`(`mod`(k,2)=0, binomial(n-k/2, k/2), 0), k = 0..n), n = 0..15); # G. C. Greubel, Jan 01 2020

Join[{0}, Table[If[EvenQ[k], Binomial[n-k/2, k/2], 0], {n,0,15}, {k,0,n} ]//Flatten] (* G. C. Greubel, Jan 01 2020 *)

row(n,d=sqrt(1+x^2/4+O(x^n))) = Vec(if(n,Pol(((x/2+d)^n-(x/2-d)^n)/d)>>1)) \\ M. F. Hasler, Dec 07 2011, edited Jul 05 2021

@CachedFunction
def T(n, k):
    if (k%2==0): return binomial(n-k/2, k/2)
    else: return 0
[0]+flatten([[T(n, k) for k in (0..n)] for n in (0..15)]) # G. C. Greubel, Jan 01 2020

P(n,x) = x*P(n-1, x) + P(n-2, x), where P(0,x)=0 and P(1,x)=1.

T(n,k) = T(n-1, k) + T(n-2, k-2) for n>=2. - Philippe Deléham, Dec 08 2013

A162514 Triangle of coefficients of polynomials defined by the Binet form P(n,x) = U^n + L^n, where U = (x + d)/2, L = (x - d)/2, d = (4 + x^2)^(1/2). Decreasing powers of x.

Original entry on oeis.org

2, 1, 0, 1, 0, 2, 1, 0, 3, 0, 1, 0, 4, 0, 2, 1, 0, 5, 0, 5, 0, 1, 0, 6, 0, 9, 0, 2, 1, 0, 7, 0, 14, 0, 7, 0, 1, 0, 8, 0, 20, 0, 16, 0, 2, 1, 0, 9, 0, 27, 0, 30, 0, 9, 0, 1, 0, 10, 0, 35, 0, 50, 0, 25, 0, 2, 1, 0, 11, 0, 44, 0, 77, 0, 55, 0, 11, 0, 1, 0, 12, 0, 54, 0, 112, 0, 105, 0, 36, 0, 2, 1, 0, 13, 0

Offset: 0

Clark Kimberling, Jul 05 2009

For a signed version of this triangle corresponding to the row reversed version of the triangle A127672 see A244422. - Wolfdieter Lang, Aug 07 2014

The row reversed triangle is A114525. - Paolo Bonzini, Jun 23 2016

			Triangle begins
   2;  == 2
   1, 0;  == x + 0
   1, 0,  2;  == x^2 + 2
   1, 0,  3, 0;  == x^3 + 3*x + 0
   1, 0,  4, 0,  2;
   1, 0,  5, 0,  5, 0;
   1, 0,  6, 0,  9, 0,  2;
   1, 0,  7, 0, 14, 0,  7, 0;
   1, 0,  8, 0, 20, 0, 16, 0,  2;
   1, 0,  9, 0, 27, 0, 30, 0,  9, 0;
   1, 0, 10, 0, 35, 0, 50, 0, 25, 0, 2;
   ...
From _Wolfdieter Lang_, Aug 07 2014: (Start)
The row polynomials R(n, x) are:
  R(0, x) = 2, R(1, x) = 1 =   x*P(1,1/x),  R(2, x) = 1 + 2*x^2 = x^2*P(2,1/x), R(3, x) = 1 + 3*x^2 = x^3*P(3,1/x), ...
(End)

G. C. Greubel, Rows n=0..100 of triangle, flattened

Cf. A000032, A114525, A162515, A162516, A162517.

Table[Reverse[CoefficientList[LucasL[n, x], x]], {n, 0, 12}]//Flatten  (* G. C. Greubel, Nov 05 2018 *)

P(n)=
{
    local(U, L, d, r, x);
    if ( n<0, return(0) );
    x = 'x+O('x^(n+1));
    d=(4 + x^2)^(1/2);
    U=(x+d)/2;  L=(x-d)/2;
    r = U^n+L^n;
    r = truncate(r);
    return( r );
}
for (n=0, 10, print(Vec(P(n))) ); /* show triangle */
/* Joerg Arndt, Jul 24 2011 */

P(n,x) = x*P(n-1,x) + P(n-2,x) for n >= 2, P(0,x) = 2, P(1,x) = x.
From Wolfdieter Lang, Aug 07 2014: (Start)
T(n,m) = [x^(n-m)] P(n,x), m = 0, 1, ..., n and  n >= 0.
G.f. of polynomials P(n,x): (2 - x*z)/(1 - x*z - z^2).
G.f. of row polynomials R(n,x) = Sum_{m=0..n} T(n,m)*x^m:  (2 - z)/(1 - z - (x*z)^2) (rows for P(n,x) reversed).
(End)
For n > 0, T(n,2*m+1) = 0, T(n,2*m) = A034807(n,m). - Paolo Bonzini, Jun 23 2016

Name clarified by Wolfdieter Lang, Aug 07 2014

A162516 Triangle of coefficients of polynomials defined by Binet form: P(n,x) = ((x+d)^n + (x-d)^n)/2, where d=sqrt(x+4).

Original entry on oeis.org

1, 1, 0, 1, 1, 4, 1, 3, 12, 0, 1, 6, 25, 8, 16, 1, 10, 45, 40, 80, 0, 1, 15, 75, 121, 252, 48, 64, 1, 21, 119, 287, 644, 336, 448, 0, 1, 28, 182, 588, 1457, 1360, 1888, 256, 256, 1, 36, 270, 1092, 3033, 4176, 6240, 2304, 2304, 0, 1, 45, 390, 1890, 5925, 10801, 17780, 11680, 12160, 1280, 1024

Offset: 0

Clark Kimberling, Jul 05 2009

			First six rows:
  1;
  1,  0;
  1,  1,  4;
  1,  3, 12,  0;
  1,  6, 25,  8, 16;
  1, 10, 48, 40, 80, 0;

G. C. Greubel, Rows n = 0..100 of triangle, flattened

Cf. A162514, A162515, A162517.
Cf. A000217, A026150, A084057, A125818, A199572.
For fixed k, the sequences P(n,k), for n=1,2,3,4,5, are A084057, A084059, A146963, A081342, A081343, respectively.

m:=12;
p:= func< n,x | ((x+Sqrt(x+4))^n + (x-Sqrt(x+4))^n)/2 >;
R:=PowerSeriesRing(Rationals(), m+1);
T:= func< n,k | Coefficient(R!( p(n,x) ), n-k) >;
[T(n,k): k in [0..n], n in [0..m]]; // G. C. Greubel, Jul 09 2023

P[n_, x_]:= P[n, x]= ((x+Sqrt[x+4])^n + (x-Sqrt[x+4])^n)/2;
T[n_, k_]:= Coefficient[Series[P[n, x], {x,0,n-k+1}], x, n-k];
Table[T[n, k], {n,0,12}, {k,0,n}]//Flatten (* G. C. Greubel, Jan 08 2020; Jul 09 2023 *)

def p(n,x): return ((x+sqrt(x+4))^n + (x-sqrt(x+4))^n)/2
def T(n,k):
    P. = PowerSeriesRing(QQ)
    return P( p(n,x) ).list()[n-k]
flatten([[T(n,k) for k in range(n+1)] for n in range(13)]) # G. C. Greubel, Jul 09 2023

P(n,x) = 2*x*P(n-1,x) - (x^2 -x -4)*P(n-2,x).
From G. C. Greubel, Jul 09 2023: (Start)
T(n, k) = [x^(n-k)] ( ((x+sqrt(x+4))^n + (x-sqrt(x+4))^n)/2 ).
T(n, 1) = A000217(n-1), n >= 1.
T(n, n) = A199572(n).
Sum_{k=0..n} T(n, k) = A084057(n).
Sum_{k=0..n} 2^k*T(n, k) = A125818(n).
Sum_{k=0..n} (-1)^k*T(n, k) = A026150(n).
Sum_{k=0..n} (-2)^k*T(n, k) = A133343(n). (End)

A192382 Coefficient of x in the reduction by x^2 -> x+2 of the polynomial p(n,x) defined below in Comments.

Original entry on oeis.org

0, 2, 4, 24, 80, 352, 1344, 5504, 21760, 87552, 349184, 1398784, 5591040, 22372352, 89473024, 357924864, 1431633920, 5726666752, 22906404864, 91626143744, 366503526400, 1466016202752, 5864060616704, 23456250855424, 93824986644480

Offset: 1

Clark Kimberling, Jun 30 2011

nonn

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

			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) = 2 + x + 3*x^2 -> 8 + 4*x.
  p(3, x) = 8*x + 4*x^2 + 4*x^3 -> 16 + 24*x.
  p(4, x) = 4 + 4*x + 21*x^2 + 10*x^3 + 5*x^4 -> 96 + 80*x.
From these, read A083086 = (1, 0, 9, 16, 96, ...) and A192382 =(0, 2, 4, 24, 80, ...).

G. C. Greubel, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, signature (2,8).

Cf. A003683, A083086, A162517, A192232.

[(4^(n-1) - (-2)^(n-1))/3: n in [1..40]]; // G. C. Greubel, Feb 19 2023

seq(4^n*(1-(-1/2)^n)/3, n=0..24); # Peter Luschny, Oct 02 2019

q[x_]:= x+2; d= Sqrt[x+2];
p[n_, x_]:= ((x+d)^n - (x-d)^n)/(2 d); (* suggested by A162517 *)
Table[Expand[p[n, x]], {n, 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,30}];
Table[Coefficient[Part[t,n], x, 0], {n,30}] (* abs value of A083086 *)
Table[Coefficient[Part[t,n], x, 1], {n,30}] (* 2*A003683 *)
Table[Coefficient[Part[t,n]/2, x, 1], {n,30}] (* A003683 *)
LinearRecurrence[{2,8}, {0,2}, 40] (* G. C. Greubel, Feb 19 2023 *)

[(4^(n-1) - (-2)^(n-1))/3 for n in range(1,41)] # G. C. Greubel, Feb 19 2023

Conjectures from Colin Barker, May 12 2014: (Start)
a(n) = 2^(n-2)*(2*(-1)^n + 2^n)/3 = 2*A003683(n-1).
a(n) = 2*a(n-1) + 8*a(n-2).
G.f.: 2*x^2 / ((1+2*x)*(1-4*x)). (End).
a(n) = 4^n*(1 - (-1/2)^n)/3. - Peter Luschny, Oct 02 2019
E.g.f: (1/3)*(2 + exp(2*x))*(sinh(x))^2. - G. C. Greubel, Feb 19 2023

A163762 Triangle of coefficients of polynomials H(n,x)=(U^n+L^n)/2+(U^n-L^n)/(2d), where U=x+d, L=x-d, d=(x+4)^(1/2).

Original entry on oeis.org

1, 1, 1, 1, 3, 4, 1, 6, 13, 4, 1, 10, 29, 24, 16, 1, 15, 55, 81, 88, 16, 1, 21, 95, 207, 300, 144, 64, 1, 28, 154, 448, 813, 684, 496, 64, 1, 36, 238, 868, 1913, 2352, 2272, 768, 256, 1, 45, 354, 1554, 4077, 6625, 7984, 4704, 2560, 256, 1, 55, 510, 2622, 8061, 16283

Offset: 1

Clark Kimberling, Aug 04 2009

H(n,x)=P(n,x)+Q(n,x), where P and Q are given by A162516, A162517.
H(n,0)=4^Floor(n/2) for n=0,1,2,...
H(n,1)=A063727(n); row sums
(Column 2)=A000217 (triangular numbers)

			First six rows:
1
1...1
1...3...4
1...6..13...4
1..10..29..24..16
1..15..55..81..88..16
Row 6 represents x^5+15*x^4+55*x^3+81*x^2+88*x+16.

A000217, A063727, A162516, A162517.

H(n,x)=2*x*H(n-1,x)-(x^2-x-4)*H(n-2,x), where H(0,x)=1, H(1,x)=x+1.

H(n,x)=(1+1/d)*U^n+(1-1/d)*L^n, where U=x+d, L=x-d, d=(x+4)^(1/2).

A192376 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

1, 0, 7, 16, 73, 256, 975, 3616, 13521, 50432, 188247, 702512, 2621849, 9784832, 36517535, 136285248, 508623521, 1898208768, 7084211623, 26438637648, 98670339049, 368242718464, 1374300534895, 5128959421024, 19141537149297, 71437189176064

Offset: 1

Clark Kimberling, Jun 29 2011

nonn

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

			The first five polynomials p(n,x) and their reductions are as follows:
p(0,x)=1 -> 1
p(1,x)=2x -> 2x
p(2,x)=4+x+3x^2 -> 7+4x
p(3,x)=16x+4x^2+4x^3 -> 16+20x
p(4,x)=16+8x+41x^2+10x^3+5x^4 -> 73+68x.
From these, read A192376=(1,0,7,16,73,...) and A192377=(0,2,4,20,68,...).

Cf. A192232, A192377, A192378.

q[x_] := x + 2; d = Sqrt[x + 1];
p[n_, x_] := ((x + d)^n - (x - d)^ n )/(2 d)   (* Cf. A162517 *)
Table[Expand[p[n, x]], {n, 1, 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, 1,  30}]
Table[Coefficient[Part[t, n], x, 0], {n, 1, 30}] (* A192376 *)
Table[Coefficient[Part[t, n], x, 1], {n, 1, 30}] (* A192377 *)
Table[Coefficient[Part[t, n]/2, x, 1], {n, 1, 30}] (* A192378 *)

Conjecture: a(n) = 2*a(n-1)+6*a(n-2)+2*a(n-3)-a(n-4). G.f.: x*(x-1)^2 / ((x+1)^2*(x^2-4*x+1)). - Colin Barker, May 11 2014

A192379 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, 0, 5, 8, 45, 128, 505, 1680, 6089, 21120, 74909, 262680, 926485, 3258112, 11474865, 40382752, 142171985, 500432640, 1761656821, 6201182760, 21829269181, 76841888640, 270495370025, 952182350768, 3351823875225, 11798909226368

Offset: 1

Clark Kimberling, Jun 29 2011

nonn

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

			The first five polynomials p(n,x) and their reductions are as follows:
p(0,x)=1 -> 1
p(1,x)=2x -> 2x
p(2,x)=2+x+3x^2 -> 5+4x
p(3,x)=8x+4x^2+4x^3 -> 8+20x
p(4,x)=4+4x+21x^2+10x^3+5x^4 -> 45+60x.
From these, read A192379=(1,0,5,8,45,...) and A192380=(0,2,4,20,60,...).

Cf. A192232, A192380, A192381.

q[x_] := x + 1; d = Sqrt[x + 2];
p[n_, x_] := ((x + d)^n - (x - d)^n )/(2 d)   (* Cf. A162517 *)
Table[Expand[p[n, x]], {n, 1, 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, 1, 30}]
Table[Coefficient[Part[t, n], x, 0], {n, 1, 30}]   (* A192379 *)
Table[Coefficient[Part[t, n], x, 1], {n, 1, 30}]   (* A192380 *)
Table[Coefficient[Part[t, n]/2, x, 1], {n, 1, 30}]   (* A192381 *)

Conjecture: a(n) = 2*a(n-1)+6*a(n-2)-2*a(n-3)-a(n-4). G.f.: -x*(x^2+2*x-1) / (x^4+2*x^3-6*x^2-2*x+1). - Colin Barker, May 11 2014

A192383 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, 0, 6, 8, 60, 160, 744, 2496, 10064, 36480, 140512, 522624, 1983168, 7439360, 28091520, 105674752, 398391552, 1500057600, 5652182528, 21288560640, 80200784896, 302101094400, 1138045495296, 4286942363648, 16149041172480, 60833034895360

Offset: 1

Clark Kimberling, Jun 30 2011

nonn

The polynomial p(n,x) is defined by ((x+d)^n - (x-d)^n)/(2*d), where d=sqrt(x+3). For an introduction to reductions of polynomials by substitutions such as x^2 -> x+1, see A192232.

			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 -> 6 + 4*x
  p(3, x) =    12*x +  4*x^2 +  4*x^3 -> 8 + 24*x
  p(4, x) = 9 + 6*x + 31*x^2 + 10*x^3 + 5*x^4 -> 60 + 72*x.
From these, read A192383 = (1, 0, 6, 8, 60, ...) and A192384 = (0, 2, 4, 24, 72, ...).

G. C. Greubel, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, signature (2,8,-4,-4).

Cf. A083084, A192232.

Cf. A192384, A192385.

R:=PowerSeriesRing(Integers(), 41);
Coefficients(R!( x*(1-2*x-2*x^2)/(1-2*x-8*x^2+4*x^3+4*x^4) )) // G. C. Greubel, Jul 10 2023

q[x_]:= x+1; d= Sqrt[x+3];
p[n_, x_]:= ((x+d)^n - (x-d)^n)/(2*d); (* suggested by A162517 *)
Table[Expand[p[n, x]], {n,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, 1, 30}];
Table[Coefficient[Part[t, n], x, 0], {n,30}] (* A192383 *)
Table[Coefficient[Part[t, n], x, 1], {n,30}] (* A192384 *)
Table[Coefficient[Part[t, n]/2, x, 1], {n,30}] (* A192385 *)
LinearRecurrence[{2,8,-4,-4}, {1,0,6,8}, 40] (* G. C. Greubel, Jul 10 2023 *)

@CachedFunction
def a(n): # a = A192383
    if (n<5): return (0,1,0,6,8)[n]
    else: return 2*a(n-1) +8*a(n-2) -4*a(n-3) -4*a(n-4)
[a(n) for n in range(1,41)] # G. C. Greubel, Jul 10 2023

From Colin Barker, May 11 2014: (Start)
a(n) = 2*a(n-1) + 8*a(n-2) - 4*a(n-3) - 4*a(n-4).
G.f.: x*(1-2*x-2*x^2)/(1-2*x-8*x^2+4*x^3+4*x^4). (End)
From G. C. Greubel, Jul 10 2023: (Start)
T(n, k) = [x^k] ((x+sqrt(x+3))^n - (x-sqrt(x+3))^n)/(2*sqrt(x+3)).
a(n) = Sum_{k=0..n-1} T(n, k)*Fibonacci(k-1). (End)

cp's OEIS Frontend

A192373 Constant term in the reduction of the polynomial p(n,x) defined at A162517 and below in Comments.

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A192374 Coefficient of x in the reduction of the polynomial p(n,x) defined at A162517 and below in Comments.

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A162515 Triangle of coefficients of polynomials defined by Binet form: P(n,x) = (U^n - L^n)/d, where U = (x + d)/2, L = (x - d)/2, d = sqrt(x^2 + 4).

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A162514 Triangle of coefficients of polynomials defined by the Binet form P(n,x) = U^n + L^n, where U = (x + d)/2, L = (x - d)/2, d = (4 + x^2)^(1/2). Decreasing powers of x.

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A162516 Triangle of coefficients of polynomials defined by Binet form: P(n,x) = ((x+d)^n + (x-d)^n)/2, where d=sqrt(x+4).

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A192382 Coefficient of x in the reduction by x^2 -> x+2 of the polynomial p(n,x) defined below in Comments.

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A163762 Triangle of coefficients of polynomials H(n,x)=(U^n+L^n)/2+(U^n-L^n)/(2d), where U=x+d, L=x-d, d=(x+4)^(1/2).

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