A192879 -id:A192879 - cp's OEIS frontend

A192872 Constant term in the reduction by (x^2 -> x+1) of the polynomial p(n,x) given in Comments.

1, 0, 3, 4, 13, 30, 81, 208, 547, 1428, 3741, 9790, 25633, 67104, 175683, 459940, 1204141, 3152478, 8253297, 21607408, 56568931, 148099380, 387729213, 1015088254, 2657535553, 6957518400

Offset: 0

Clark Kimberling, Jul 11 2011

The polynomial p(n,x) is defined by p(0,x)=1, p(1,x)=x, and p(n,x) = x*p(n-1,x) + (x^2)*p(n-1,x) + 1. 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*p(n-1,x) + e*(x^2)*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, formally, linear recurrence sequences of order 5.  The second Mathematica program below shows initial terms and the recurrence coefficients, which are too long to be included here, which imply these properties:
(1) The numbers a,b,c,f affect initial terms but not the recurrence coefficients, which depend only on u,v,d,e.
(2) If v=0 or e=0, the order of recurrence is <= 3.
(3) If v=0 and e=0, the order of recurrence is 2, and the coefficients are 1+d*u and d*u.
(See A192904 for similar results for other p(n,x).)
...
Examples:
u v a b c d e f seq h.....seq k
1 1 1 2 0 1 1 0 -A121646..A059929
1 1 1 3 0 1 1 0 A128533...A081714
1 1 2 1 0 1 1 0 A081714...A001906
1 1 1 1 1 1 1 0 A000045...A001906
1 1 2 1 1 1 1 0 A129905...A192879
1 1 1 2 1 1 1 0 A061646...A079472
1 1 1 1 0 1 1 1 A192872...A192873
1 1 1 1 1 2 1 1 A192874...A192875
1 1 1 1 1 2 1 1 A192876...A192877
1 1 1 1 1 1 2 1 A192880...A192882
1 1 1 1 1 1 1 1 A166536...A064831
The terms of several of these sequences are products of Fibonacci numbers (A000045), or Fibonacci numbers and Lucas numbers (A000032).

			The coefficients in all the polynomials p(n,x) are Fibonacci numbers (A000045).  The first six and their reductions:
p(0,x) = 1 -> 1
p(1,x) = x -> x
p(2,x) = 1 + 2*x^2 -> 3 + 2*x
p(3,x) = 1 + x + 3*x^3 -> 4 + 7*x
p(4,x) = 1 + x + 2*x^2 + 5*x^4 -> 13 + 18*x
p(5,x) = 1 + x + 2*x^2 + 3*x^3 + 8*x^5 -> 30 + 49*x

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

Cf. A192232, A192744, A192873, A192908 (sums of adjacent terms).

a:=[1,0,3,4];; for n in [5..30] do a[n]:=3*a[n-1]-3*a[n-3]+a[n-4]; od; a; # G. C. Greubel, Jan 06 2019

m:=30; R:=PowerSeriesRing(Integers(), m); Coefficients(R!( (2*x-1)*(x^2-x+1)/((x-1)*(1+x)*(x^2-3*x +1)) )); // G. C. Greubel, Jan 06 2019

(* First program *)
q = x^2; s = x + 1; z = 26;
p[0, x_] := 1; p[1, x_] := x;
p[n_, x_] := p[n - 1, x]*x + p[n - 2, x]*x^2 + 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}] (* A192872 *)
u2 = Table[Coefficient[Part[t, n], x, 1], {n, 1, z}] (* A192873 *)
(* End of 1st program *)
(* ******************************************** *)
(* Second program: much more general *)
(* u = 1; v = 1; a = 1; b = 1; c = 0; d = 1; e = 1; f = 1; Nine degrees of freedom for user; shown values generate A192872. *)
q = x^2; s = u*x + v; z = 11;
(* will apply reduction (x^2 -> u*x+v) to p(n,x) *)
p[0, x_] := a; p[1, x_] := b*x + c;
(* initial values of polynomial sequence p(n,x) *)
p[n_, x_] := d*x*p[n - 1, x] + e*(x^2)*p[n - 2, x] + f;
(* recurrence for p(n,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}];
u2 = Table[Coefficient[Part[t, n], x, 1], {n, 1, z}];
Simplify[FindLinearRecurrence[u1]] (* for 0-sequence *)
Simplify[FindLinearRecurrence[u2]] (* for 1-sequence *)
u1 = Table[Coefficient[Part[t, n], x, 0], {n, 1, 4}]
(* initial values for 0-sequence *)
u2 = Table[Coefficient[Part[t, n], x, 1], {n, 1, 4}]
(* initial values for 1-sequence *)
LinearRecurrence[{3,0,-3,1},{1,0,3,4},26] (* Ray Chandler, Aug 02 2015 *)

my(x='x+O('x^30)); Vec((2*x-1)*(x^2-x+1)/((x-1)*(1+x)*(x^2-3*x +1))) \\ G. C. Greubel, Jan 06 2019

((2*x-1)*(x^2-x+1)/((x-1)*(1+x)*(x^2-3*x +1))).series(x, 30).coefficients(x, sparse=False) # G. C. Greubel, Jan 06 2019

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

G.f.: (2*x-1)*(x^2-x+1) / ( (x-1)*(1+x)*(x^2-3*x+1) ). - R. J. Mathar, Oct 26 2011

A192878 Constant term in the reduction by (x^2 -> x + 1) of the polynomial p(n,x) given in Comments.

Original entry on oeis.org

3, 0, 4, 5, 18, 42, 115, 296, 780, 2037, 5338, 13970, 36579, 95760, 250708, 656357, 1718370, 4498746, 11777875, 30834872, 80726748, 211345365, 553309354, 1448582690, 3792438723, 9928733472, 25993761700, 68052551621, 178163893170, 466439127882, 1221153490483

Offset: 0

Clark Kimberling, Jul 11 2011

The polynomial p(n,x) is defined by p(0,x) = 1, p(1,x) = x + 1, and p(n,x) = x*p(n-1,x) + 2*(x^2)*p(n-1,x) + 1. See A192872.

For n>0, also the coefficient of x in the reduction x^2 -> x + 1 of the polynomial A000285(n-1)*x^(n-1). - R. J. Mathar, Jul 12 2011

			The first six polynomials and reductions:
p(0,x) = 3 -> 3
p(1,x) = x -> x
p(2,x) = 4*x^2 -> 4 + 4*x
p(3,x) = 5*x^3 -> 5 + 10*x
p(4,x) = 9*x^4 -> 18 + 27*x
p(5,x) = 14*x^5 -> 42 + 27*x
In general, p(n,x) = A104449(n)*x^n -> A192878(n) + A192879(n)*x.

Colin Barker, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (2,2,-1).

Cf. A000285, A192872, A192879.

a:=[3,0,4];; for n in [4..30] do a[n]:=2*a[n-1]+2*a[n-2]-a[n-3]; od; a; # G. C. Greubel, Jan 08 2019

m:=30; R:=PowerSeriesRing(Integers(), m); Coefficients(R!( (3-2*x^2-6*x)/((1+x)*(x^2-3*x+1)) )); // G. C. Greubel, Jan 08 2019

q = x^2; s = x + 1; z = 25;
p[0, x_] := 3; p[1, x_] := x;
p[n_, x_] := p[n - 1, x]*x + p[n - 2, x]*x^2;
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}]  (* A192878 *)
u2 = Table[Coefficient[Part[t, n], x, 1], {n, 1, z}]  (* A192879 *)
FindLinearRecurrence[u1]
FindLinearRecurrence[u2]
LinearRecurrence[{2,2,-1}, {3,0,4}, 30] (* G. C. Greubel, Jan 08 2019 *)

a(n) = round((2^(-n)*(7*(-2)^n-(-4+sqrt(5))*(3+sqrt(5))^n+(3-sqrt(5))^n*(4+sqrt(5))))/5) \\ Colin Barker, Sep 29 2016

Vec((3-2*x^2-6*x)/((1+x)*(x^2-3*x+1)) + O(x^40)) \\ Colin Barker, Sep 29 2016

((3-2*x^2-6*x)/((1+x)*(x^2-3*x+1))).series(x, 30).coefficients(x, sparse=False) # G. C. Greubel, Jan 08 2019

a(n) = 2*a(n-1) + 2*a(n-2) - a(n-3).
G.f.: ( 3 - 2*x^2 - 6*x ) / ( (1+x)*(1 - 3*x + x^2) ). - R. J. Mathar, May 07 2014
a(n) = (2^(-n)*(7*(-2)^n - (-4+sqrt(5))*(3+sqrt(5))^n + (3-sqrt(5))^n*(4+sqrt(5))))/5. - Colin Barker, Sep 29 2016

A192921 Constant term in the reduction by (x^2->x+1) of the polynomial p(n,x) defined below at Comments.

Original entry on oeis.org

1, 2, 2, 7, 16, 44, 113, 298, 778, 2039, 5336, 13972, 36577, 95762, 250706, 656359, 1718368, 4498748, 11777873, 30834874, 80726746, 211345367, 553309352, 1448582692, 3792438721, 9928733474, 25993761698, 68052551623, 178163893168, 466439127884, 1221153490481

Offset: 0

Clark Kimberling, Jul 12 2011

The titular polynomial is defined by p(n,x) = x*p(n-1,x) +(x^2)*p(n-2,x), with p(0,x)=1, p(1,x)=1+x^2. For discussions of polynomial reduction, see A192232, A192744, and A192872.

			The coefficients in the polynomials p(n,x) are Fibonacci numbers.  The first seven and their reductions:
...
1 -> 1
1 + x^2 -> 2 + x
x + x^2 + x^3 -> 2 + 4*x
2*x^2 + x^3 + 2*x^4 -> 7 + 10*x
3*x^3 + 2*x^4 + 3*x^5 -> 16 + 27*x
5*x^4 + 3*x^5 + 5*x^6 -> 44 + 70*x
8*x^5 + 5*x^6 + 8*x^7 -> 113 + 184*x,
so that A192921=(1,2,2,7,16,44,113,...).

Alois P. Heinz, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (2,2,-1).

Cf. A192232, A192744, A192872.

List([0..30], n -> Fibonacci(n-2)^2 +Fibonacci(n)*Fibonacci(n+1)); # G. C. Greubel, Feb 06 2019

[Fibonacci(n-2)^2 + Fibonacci(n)*Fibonacci(n+1): n in [0..30]]; // G. C. Greubel, Feb 06 2019

a:= n-> (<<0|1|0>, <0|0|1>, <-1|2|2>>^n. <<1,2,2>>)[1,1]:
seq(a(n), n=0..30);  # Alois P. Heinz, Sep 28 2016

q = x^2; s = x + 1; z = 28;
p[0, x_] := 1; p[1, x_] := x^2 + 1;
p[n_, x_] := p[n - 1, x]*x + p[n - 2, x]*x^2;
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}]
  (* A192921 *)
u2 = Table[Coefficient[Part[t, n], x, 1], {n, 1, z}]
  (* A192879 *)
LinearRecurrence[{2,2,-1}, {1,2,2}, 30] (* G. C. Greubel, Feb 06 2019 *)

a(n) = round((2^(-n)*(-3*(-2)^n-(-4+sqrt(5))*(3+sqrt(5))^n+(3-sqrt(5))^n*(4+sqrt(5))))/5) \\ Colin Barker, Oct 01 2016

Vec(-(2*x-1)*(1+2*x)/((1+x)*(x^2-3*x+1)) + O(x^40)) \\ Colin Barker, Oct 01 2016

{a(n) = fibonacci(n-2)^2 +fibonacci(n)*fibonacci(n+1)}; \\ G. C. Greubel, Feb 06 2019

[fibonacci(n-2)^2 +fibonacci(n)*fibonacci(n+1) for n in range(30)] # G. C. Greubel, Feb 06 2019

a(n) = 2*a(n-1) + 2*a(n-2) - a(n-3).
G.f.: (1-2*x)*(1+2*x) / ( (1+x)*(1-3*x+x^2) ). - R. J. Mathar, May 08 2014
a(n) = A059929(n-1) + 2*A059929(n-2). - R. J. Mathar, May 08 2014
a(n) = F(n-4)*F(n) + F(n-1)*F(n+2), where F(-4)=-3, F(-3)=2, F(-2)=-1, F(-1)=1. - Bruno Berselli, Nov 03 2015
a(n) = (2^(-n)*(-3*(-2)^n-(-4+sqrt(5))*(3+sqrt(5))^n+(3-sqrt(5))^n*(4+sqrt(5))))/5. - Colin Barker, Oct 01 2016

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A192872 Constant term in the reduction by (x^2 -> x+1) of the polynomial p(n,x) given in Comments.

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A192878 Constant term in the reduction by (x^2 -> x + 1) of the polynomial p(n,x) given in Comments.

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A192921 Constant term in the reduction by (x^2->x+1) of the polynomial p(n,x) defined below at Comments.

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