A192232 -id:A192232 - cp's OEIS frontend

A192904 Constant term in the reduction by (x^2 -> x + 1) of the polynomial p(n,x) defined below at Comments.

1, 0, 1, 5, 16, 49, 153, 480, 1505, 4717, 14784, 46337, 145233, 455200, 1426721, 4471733, 14015632, 43928817, 137684905, 431542080, 1352570689, 4239325789, 13287204352, 41645725825, 130529073953, 409113752000, 1282274186177

Offset: 0

Clark Kimberling, Jul 12 2011

The titular polynomial is defined by p(n,x) = (x^2)*p(n-1,x) + x*p(n-2,x), with p(0,x) = 1, p(1,x) = x.  The resulting sequence typifies a general class which we shall describe here. Suppose that u,v,a,b,c,d,e,f are numbers used to define these polynomials:
...
q(x) = x^2
s(x) = u*x + v
p(0,x) = a, p(1,x) = b*x + c
p(n,x) = d*(x^2)*p(n-1,x) + e*x*p(n-2,x) + f.
...
We shall assume that u is not 0 and that {d,e} is not {0}.  The reduction of p(n,x) by the repeated substitution q(x) -> s(x), as defined and described at A192232 and A192744, has the form h(n) + k(n)*x.  The numerical sequences h and k are linear recurrence sequences, formally of order 5.  The Mathematica program below, with first line deleted, shows initial terms and recurrence coefficients, which imply these properties:
(1)  the recurrence coefficients depend only on u,v,d,e; the parameters a,b,c,f affect only the initial terms.
(2)  if e=0 or v=0, the order of recurrence is <= 3;
(3)  if e=0 and v=0, the recurrence coefficients are 1+d*u^2 and -d*u^2 (cf. similar results at A192872).
...
Examples:
u v a b c d e f... seq h.....seq k
1 1 1 1 1 1 0 0... A001906..A001519
1 1 1 1 0 0 1 0... A103609..A193609
1 1 1 1 0 1 1 0... A192904..A192905
1 1 1 1 1 1 0 0... A001519..A001906
1 1 1 1 1 1 1 0... A192907..A192907
1 1 1 1 1 1 0 1... A192908..A069403
1 1 1 1 1 1 1 1... A192909..A192910
The terms of these sequences involve Fibonacci numbers, F(n)=A000045(n); e.g.,
A001906: even-indexed F(n)
A001519: odd-indexed F(n)
A103609: (1,1,1,1,2,2,3,3,5,5,8,8,...)

			The first six polynomials and reductions:
1 -> 1
x -> x
x + x^3 -> 1 + 3*x
x^2 + x^3 + x^5 -> 5 + 8*x
x^2 + 2*x^4 + x^5 + x^7 -> 16 + 25*x
x^3 + 2*x^4 + 3*x^6 + x^7 + x^9 -> 49 + 79*x, so that
A192904 = (1,0,1,5,16,49,...) and
A192905 = (0,1,3,8,25,79,...)

G. C. Greubel, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (3,0,1,1).

Cf. A192232, A192744, A192905, A192872.

a:=[1,0,1,5];; for n in [5..40] do a[n]:=3*a[n-1]+a[n-3]+a[n-4]; od; a; # G. C. Greubel, Jan 10 2019

m:=40; R:=PowerSeriesRing(Integers(), m); Coefficients(R!( (1-x)*(1-2*x-x^2)/(1-3*x-x^3-x^4) )); // G. C. Greubel, Jan 10 2019

(* To obtain general results, delete the next line. *)
u = 1; v = 1; a = 1; b = 1; c = 0; d = 1; e = 1; f = 0;
q = x^2; s = u*x + v; z = 24;
p[0, x_] := a; p[1, x_] := b*x + c;
p[n_, x_] :=  d*(x^2)*p[n - 1, x] + e*x*p[n - 2, x] + f;
Table[Expand[p[n, x]], {n, 0, 8}]
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}];
u0 = Table[Coefficient[Part[t, n], x, 0], {n, 1, z}] (* A192904 *)
u1 = Table[Coefficient[Part[t, n], x, 1], {n, 1, z}] (* A192905 *)
Simplify[FindLinearRecurrence[u0]] (* recurrence for 0-sequence *)
Simplify[FindLinearRecurrence[u1]] (* recurrence for 1-sequence *)
LinearRecurrence[{3,0,1,1}, {1,0,1,5}, 40] (* G. C. Greubel, Jan 10 2019 *)

my(x='x+O('x^40)); Vec((1-x)*(1-2*x-x^2)/(1-3*x-x^3-x^4)) \\ G. C. Greubel, Jan 10 2019

((1-x)*(1-2*x-x^2)/(1-3*x-x^3-x^4)).series(x, 40).coefficients(x, sparse=False) # G. C. Greubel, Jan 10 2019

a(n) = 3*a(n-1) + a(n-3) + a(n-4).

G.f.: (1-x)*(1-2*x-x^2)/(1-3*x-x^3-x^4). - Colin Barker, Aug 31 2012

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

A192425 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, 1, 1, 6, 9, 31, 60, 169, 369, 954, 2201, 5479, 12960, 31721, 75881, 184326, 443169, 1072871, 2585340, 6249329, 15074649, 36413754, 87877681, 212208719, 512231040, 1236774481, 2985612241, 7208270406, 17401713849, 42012408751

Offset: 0

Clark Kimberling, Jun 30 2011

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.

			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*A192424(n) = (2, 0, 4, 2, 16, ...) and a(n) = (0, 1, 1, 6, 9, ...).

G. C. Greubel, Table of n, a(n) for n = 0..1000
H. C. Williams and R. K. Guy, Some fourth-order linear divisibility sequences, Intl. J. Number Theory 7 (5) (2011) 1255-1277.
H. C. Williams and R. K. Guy, Some Monoapparitic Fourth Order Linear Divisibility Sequences Integers, Volume 12A (2012) The John Selfridge Memorial Volume
Index entries for linear recurrences with constant coefficients, signature (1,4,-1,-1).

Cf. A000045, A000129, A001045, A002878, A192232, A192423, A192424.

R:=PowerSeriesRing(Integers(), 40); [0] cat Coefficients(R!( x*(1+x^2)/((1+x-x^2)*(1-2*x-x^2)) )); // G. C. Greubel, Jul 12 2023

(See A192423.)
LinearRecurrence[{1,4,-1,-1}, {0,1,1,6}, 40] (* G. C. Greubel, Jul 12 2023 *)

@CachedFunction
def a(n): # a = A192425
    if (n<4): return (0,1,1,6)[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

G.f.: x*(1+x^2)/((1+x-x^2)*(1-2*x-x^2)). Colin Barker, Nov 13 2012
From Peter Bala, Mar 26 2015: (Start)
The following remarks assume the o.g.f. for this sequence is x*(1 + x^2)/((1 + x - x^2)*(1 - 2*x - x^2)).
This sequence is a fourth-order linear divisibility sequence. It is the case P1 = 1, P2 = -2, Q = -1 of the 3-parameter family of divisibility sequences found by Williams and Guy.
exp( Sum_{n >= 1} 3*a(n)*x^n/n ) = 1 + Sum_{n >= 1} 3*Pell(n)*x^n.
exp( Sum_{n >= 1} (-3)*a(n)*x^n/n ) = 1 + Sum_{n >= 1} 3*Fibonacci(n)*(-x)^n. Cf. A002878. (End)
From G. C. Greubel, Jul 12 2023: (Start)
a(n) = Sum_{j=0..n} T(n, j)*A001045(j), where T(n, k) = [x^k] ((x + sqrt(x^2+4))^n + (x - sqrt(x^2+4))^n)/2^n.
a(n) = a(n-1) + 4*a(n-2) - a(n-3) - a(n-4).
a(n) = (1/3)*( 2*A000129(n+1) - 2*A000129(n) - (-1)^n*A000032(n)). (End)

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

Clark Kimberling, Jul 09 2011

For discussions of polynomial reduction, see A192232 and A192744.

			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,...)

Index entries for linear recurrences with constant coefficients, signature (1,5,-1,-5,1,1).

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

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 *)

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

Clark Kimberling, Jul 09 2011

nonn

For discussions of polynomial reduction, see A192232 and A192744.

			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,...)

Index entries for linear recurrences with constant coefficients, signature (1,6,-1,-6,1,1).

Cf. A192744, A192232, A192616, A192772, A192778, A192779.

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 *)

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

Clark Kimberling, Jul 09 2011

For discussions of polynomial reduction, see A192232 and A192744.

			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,...)

Index entries for linear recurrences with constant coefficients, signature (1,6,-1,-6,1,1).

Cf. A192744, A192232, A192616, A192772, A192777, A192779.

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

Clark Kimberling, Jul 09 2011

For discussions of polynomial reduction, see A192232 and A192744.

			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,...)

Index entries for linear recurrences with constant coefficients, signature (1,6,-1,-6,1,1).

Cf. A192744, A192232, A192616, A192772, A192777.

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

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

Clark Kimberling, Jul 10 2011

For discussions of polynomial reduction, see A192232 and A192744.

			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,...)

Index entries for linear recurrences with constant coefficients, signature (1,3,0,-3,1,1).

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

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 *)

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]

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

A192799 Coefficient of x in the reduction of the n-th Fibonacci polynomial by x^3->x^2+2.

Original entry on oeis.org

0, 1, 0, 2, 2, 5, 12, 22, 54, 109, 242, 520, 1118, 2427, 5218, 11290, 24352, 52579, 113526, 245038, 529068, 1142087, 2465644, 5322896, 11491188, 24807721, 53555508, 115617714, 249599214, 538843277, 1163273304, 2511313222, 5421508714

Offset: 1

Clark Kimberling, Jul 10 2011

For discussions of polynomial reduction, see A192232 and A192744.

			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,...)

Index entries for linear recurrences with constant coefficients, signature (1,3,0,-3,1,1).

Cf. A192232, A192744, A192798.

Mathematica
```
(See A192798.)
```

a(n) = a(n-1)+3*a(n-2)-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+3*x^2+x-1). [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

Clark Kimberling, Jul 10 2011

For discussions of polynomial reduction, see A192232 and A192744.

			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,...)

Index entries for linear recurrences with constant coefficients, signature (1,3,0,-3,1,1).

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

Mathematica
```
(See A192798.)
```

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]

cp's OEIS Frontend

A192904 Constant term in the reduction by (x^2 -> x + 1) of the polynomial p(n,x) defined below at Comments.

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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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A192425 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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A192772 Constant term in the reduction of the n-th Fibonacci polynomial by x^3->x^2+2x+1.

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A192777 Constant term in the reduction of the n-th Fibonacci polynomial by x^3->x^2+3x+1. See Comments.

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A192778 Coefficient of x in the reduction of the n-th Fibonacci polynomial by x^3->x^2+3x+1.

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

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