A192744 -id:A192744 - cp's OEIS frontend

A192761 Coefficient of x in the reduction by x^2->x+1 of the polynomial p(n,x) defined below in Comments.

0, 1, 5, 11, 22, 40, 70, 119, 199, 329, 540, 882, 1436, 2333, 3785, 6135, 9938, 16092, 26050, 42163, 68235, 110421, 178680, 289126, 467832, 756985, 1224845, 1981859, 3206734, 5188624, 8395390, 13584047, 21979471, 35563553, 57543060, 93106650

Offset: 0

Clark Kimberling, Jul 09 2011

The titular polynomial is defined recursively by p(n,x) = x*(n-1,x) + n + 3 for n > 0, where p(0,x) = 1. For discussions of polynomial reduction, see A192232 and A192744.

Construct a triangle with T(n,0) = n*(n+1)+1 and T(n,n) = (n+1)*(n+2)/2 starting at n=0. Define the interior terms by T(r,c) = T(r-2,c-1) + T(r-1,c). The sequence of its row sums is 1, 6, 17, 39, 79, 149, 268, 467,... and the first differences of these (the sum of the terms in row(n) less those in row(n-1)) equals a(n+1). - J. M. Bergot, Mar 10 2013

Index entries for linear recurrences with constant coefficients, signature (3,-2,-1,1).

Cf. A192744, A192232, partial sums of A022318.

q = x^2; s = x + 1; z = 40;
p[0, n_] := 1; p[n_, x_] := x*p[n - 1, x] + 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}]
  (* A022318 *)
u2 = Table[Coefficient[Part[t, n], x, 1], {n, 1, z}]
  (* A192761 *)

a(n) = 3*a(n-1)-2*a(n-2)-a(n-3)+a(n-4). G.f.: x*(2*x^2-2*x-1) / ((x-1)^2*(x^2+x-1)). [Colin Barker, Dec 08 2012]

A192808 Constant term in the reduction of the polynomial (x^2 + 2)^n by x^3 -> x^2 + 2. See Comments.

Original entry on oeis.org

1, 2, 6, 26, 126, 618, 3022, 14746, 71902, 350538, 1708910, 8331130, 40615294, 198004778, 965298958, 4705957722, 22942154782, 111845982474, 545263681710, 2658231220538, 12959222223038, 63177890368490, 308000415667278, 1501542003033370

Offset: 0

Clark Kimberling, Jul 10 2011

For discussions of polynomial reduction, see A192232 and A192744.
If the reduction (x^2 + c)^n by x^3 -> x^2 + c is applied to the polynomials (x^2+c)^n for c=1 instead of c=2, the results are as follows:
A052554: constant terms,
A052529: coefficients of x,
A124820: coefficients of x^2.
Those three sequences satisfy the recurrence:
u(n) = 4*u(n-1) - 3*u(n-2) + u(n-3).

G. C. Greubel, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (7,-12,8).

Cf. A192744, A192232, A192808 - A192811.

a:=[1,2,6];; for n in [4..25] do a[n]:=7*a[n-1]-12*a[n-2]+8*a[n-3]; od; Print(a); # Muniru A Asiru, Jan 02 2019

m:=30; R:=PowerSeriesRing(Integers(), m); Coefficients(R!( (1-x)*(1-4*x)/(1-7*x+12*x^2-8*x^3) )); // G. C. Greubel, Jan 02 2019

q = x^3; s = x^2 + 2; z = 40;
p[n_, x_] := (x^2 + 2)^n;
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}] (* A192808 *)
u2 = Table[Coefficient[Part[t, n], x, 1], {n, 1, z}] (* A192809 *)
u3 = Table[Coefficient[Part[t, n], x, 2], {n, 1, z}] (* A192810 *)
uu = u2/2  (* A192811 *)
LinearRecurrence[{7,-12,8}, {1,2,6}, 50] (* G. C. Greubel, Jan 02 2019 *)

my(x='x+O('x^30)); Vec((1-x)*(1-4*x)/(1-7*x+12*x^2-8*x^3)) \\ G. C. Greubel, Jan 02 2019

((1-x)*(1-4*x)/(1-7*x+12*x^2-8*x^3)).series(x, 30).coefficients(x, sparse=False) # G. C. Greubel, Jan 02 2019

a(n) = 7*a(n-1) - 12*a(n-2) + 8*a(n-3).

G.f.: (1-x)*(1-4*x)/(1-7*x+12*x^2-8*x^3). - Colin Barker, Jul 26 2012

A192960 Constant term of the reduction by x^2 -> x+1 of the polynomial p(n,x) defined at Comments.

Original entry on oeis.org

1, 3, 7, 15, 29, 53, 93, 159, 267, 443, 729, 1193, 1945, 3163, 5135, 8327, 13493, 21853, 35381, 57271, 92691, 150003, 242737, 392785, 635569, 1028403, 1664023, 2692479, 4356557, 7049093, 11405709, 18454863, 29860635, 48315563, 78176265

Offset: 0

Clark Kimberling, Jul 13 2011

nonn

The titular polynomials are defined recursively: p(n,x) = x*p(n-1,x) + 2 + n^2, with p(0,x)=1. For an introduction to reductions of polynomials by substitutions such as x^2 -> x+1, see A192232 and A192744.

G. C. Greubel, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (3,-2,-1,1).

Cf. A000045, A192232, A192744, A192951, A192961.

F:=Fibonacci;; List([0..40], n-> 2*F(n+4)-(2*n+5)); # G. C. Greubel, Jul 12 2019

F:=Fibonacci; [2*F(n+4)-(2*n+5): n in [0..40]]; // G. C. Greubel, Jul 12 2019

(* First program *)
q = x^2; s = x + 1; z = 40;
p[0, x]:= 1;
p[n_, x_]:= x*p[n - 1, x] + n^2 + 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}] (* A192960 *)
u2 = Table[Coefficient[Part[t, n], x, 1], {n, 1, z}] (* A192961 *)
(* Second program *)
With[{F=Fibonacci}, Table[2*F[n+4]-(2*n+5), {n,0,40}]] (* G. C. Greubel, Jul 12 2019 *)

vector(40, n, n--; f=fibonacci; 2*f(n+4)-(2*n+5)) \\ G. C. Greubel, Jul 12 2019

f=fibonacci; [2*f(n+4)-(2*n+5) for n in (0..40)] # G. C. Greubel, Jul 12 2019

a(n) = 3*a(n-1) - 2*a(n-2) - a(n-3) + a(n-4).
From R. J. Mathar, May 09 2014: (Start)
G.f.: (1+x)*(1-x+x^2)/((1-x-x^2)*(1-x)^2).
a(n) - a(n-1) = A019274(n+2). (End)
a(n) = 2*Fibonacci(n+4) - (2*n + 5). - G. C. Greubel, Jul 12 2019

A192967 Constant term of the reduction by x^2 -> x+1 of the polynomial p(n,x) defined at Comments.

Original entry on oeis.org

1, 0, 2, 4, 9, 17, 31, 54, 92, 154, 255, 419, 685, 1116, 1814, 2944, 4773, 7733, 12523, 20274, 32816, 53110, 85947, 139079, 225049, 364152, 589226, 953404, 1542657, 2496089, 4038775, 6534894, 10573700, 17108626, 27682359, 44791019, 72473413, 117264468

Offset: 0

Clark Kimberling, Jul 13 2011

The titular polynomials are defined recursively: p(n,x) = x*p(n-1,x) + n(n-1)/2, with p(0,x)=1. For an introduction to reductions of polynomials by substitutions such as x^2->x+1, see A192232 and A192744.

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (3,-2,-1,1).

Cf. A000045, A192232, A192744, A192951.

List([0..40], n-> 3*Fibonacci(n+1) -n-2); # G. C. Greubel, Jul 11 2019

I:=[1, 0, 2, 4]; [n le 4 select I[n] else 3*Self(n-1)-2*Self(n-2)-Self(n-3)+Self(n-4): n in [1..40]]; // Vincenzo Librandi, Feb 20 2012

[3*Fibonacci(n+1) -n-2: n in [0..40]]; // G. C. Greubel, Jul 11 2019

q = x^2; s = x + 1; z = 40;
p[0, x]:= 1;
p[n_, x_]:= x*p[n-1, x] + n*(n-1)/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}] (* A192967 *)
u2 = Table[Coefficient[Part[t, n], x, 1], {n, 1, z}] (* A192968 *)
LinearRecurrence[{3,-2,-1,1}, {1,0,2,4}, 41] (* Vladimir Joseph Stephan Orlovsky, Feb 15 2012 *)
Table[3*Fibonacci[n+1] -n-2, {n,0,40}] (* G. C. Greubel, Jul 11 2019 *)

vector(40, n, n--; f=fibonacci; 3*f(n+1)-n-2) \\ G. C. Greubel, Jul 11 2019

[3*fibonacci(n+1) -n-2 for n in (0..40)] # G. C. Greubel, Jul 11 2019

a(0)=1, a(1)=0, for n > 1, a(n) = a(n-1) + a(n-2) + n - 1. - Alex Ratushnyak, May 10 2012
a(n) = 3*a(n-1) - 2*a(n-2) - a(n-3) + a(n-4).
G.f.: (1 -3*x +4*x^2 -x^3)/((1-x-x^2)*(1-x)^2). - R. J. Mathar, May 11 2014
a(n) = 3*Fibonacci(n+1) - n - 2. - G. C. Greubel, Jul 11 2019

A192971 Constant term of the reduction by x^2 -> x+1 of the polynomial p(n,x) defined at Comments.

Original entry on oeis.org

1, 2, 9, 21, 44, 83, 149, 258, 437, 729, 1204, 1975, 3225, 5250, 8529, 13837, 22428, 36331, 58829, 95234, 154141, 249457, 403684, 653231, 1057009, 1710338, 2767449, 4477893, 7245452, 11723459, 18969029, 30692610, 49661765, 80354505

Offset: 0

Clark Kimberling, Jul 13 2011

nonn

The titular polynomials are defined recursively: p(n,x) = x*p(n-1,x) + 2*n^2, with p(0,x)=1. For an introduction to reductions of polynomials by substitutions such as x^2 -> x+1, see A192232 and A192744.

G. C. Greubel, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (3,-2,-1,1).

Cf. A000032, A000045, A192232, A192744, A192951, A192972.

F:=Fibonacci;; List([0..40], n-> 5*F(n+3)+F(n+1) -2*(2*n+5)); # G. C. Greubel, Jul 24 2019

F:=Fibonacci; [5*F(n+3)+F(n+1) -2*(2*n+5): n in [0..40]]; // G. C. Greubel, Jul 24 2019

(* First program *)
q = x^2; s = x + 1; z = 40;
p[0, x]:= 1;
p[n_, x_]:= x*p[n-1, x] + 2*n^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}] (* A192971 *)
u2 = Table[Coefficient[Part[t, n], x, 1], {n, 1, z}] (* A192972 *)
(* Additional programs *)
With[{F = Fibonacci}, Table[5*F[n+3]+F[n+1] -2*(2*n+5), {n,0,40}]] (* G. C. Greubel, Jul 24 2019 *)

vector(40, n, n--; f=fibonacci; 5*f(n+3)+f(n+1) -2*(2*n+5)) \\ G. C. Greubel, Jul 24 2019

f=fibonacci; [5*f(n+3)+f(n+1) -2*(2*n+5) for n in (0..40)] # G. C. Greubel, Jul 24 2019

a(n) = 3*a(n-1) - 2*a(n-2) - a(n-3) + a(n-4).
G.f.: (1-x+5*x^2-x^3)/((1-x-x^2)*(1-x)^2). - R. J. Mathar, May 11 2014
a(n) = 4*Fibonacci(n+3) + Lucas(n+2) - 2*(2*n+5). - G. C. Greubel, Jul 24 2019

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

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+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,...)

Index entries for linear recurrences with constant coefficients, signature (1,4,-1,-4,1,1).

Cf. A192744, A192232, A192616.

(See A192616.)
LinearRecurrence[{1,4,-1,-4,1,1},{0,1,0,3,2,10},40] (* Harvey P. Dale, Feb 23 2021 *)

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

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+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,...)

Index entries for linear recurrences with constant coefficients, signature (1,4,-1,-4,1,1).

Cf. A192232, A192744, A192616.

Mathematica
```
(See A192616.)
```

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

Clark Kimberling, Jul 09 2011

nonn

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.

Cf. A192744, A192232, A192753.

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 *)

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

A192753 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, 1, 8, 20, 43, 82, 148, 257, 436, 728, 1203, 1974, 3224, 5249, 8528, 13836, 22427, 36330, 58828, 95233, 154140, 249456, 403683, 653230, 1057008, 1710337, 2767448, 4477892, 7245451, 11723458, 18969028, 30692609, 49661764, 80354504, 130016403

Offset: 0

Clark Kimberling, Jul 09 2011

nonn

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.

Cf. A192232, A192744, A192753.

Mathematica
```
(See A192752.)
```

Conjecture: G.f.: x*(-1-5*x+2*x^2) / ( (x^2+x-1)*(x-1)^2 ), with first differences in A192752. - R. J. Mathar, May 04 2014

A192754 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, 6, 12, 23, 40, 68, 113, 186, 304, 495, 804, 1304, 2113, 3422, 5540, 8967, 14512, 23484, 38001, 61490, 99496, 160991, 260492, 421488, 681985, 1103478, 1785468, 2888951, 4674424, 7563380, 12237809, 19801194, 32039008, 51840207, 83879220, 135719432

Offset: 0

Clark Kimberling, Jul 09 2011

The titular polynomial is defined recursively by p(n,x)=x*p(n-1,x)+5*n+1 for n>0, where p(0,x)=1. For discussions of polynomial reduction, see A192232 and A192744.

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

Cf. A192744, A192232, A192755.

p[0, n_] := 1; p[n_, x_] := x*p[n - 1, x] + 5 n + 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}]
  (* A192754 *)
u2 = Table[Coefficient[Part[t, n], x, 1], {n, 1, z}]
  (* A192755 *)
LinearRecurrence[{2, 0, -1}, {1, 6, 12}, 60] (* Vladimir Joseph Stephan Orlovsky, Feb 15 2012 *)

Conjecture: G.f.: ( 1+4*x ) / ( (x-1)*(x^2+x-1) ), partial sums of A022095. a(n) = A000071(n+3)+4*A000071(n+2). - R. J. Mathar, May 04 2014

a(n) = 8*Fibonacci(n) + 3*Lucas(n) - 5. - Greg Dresden, Oct 10 2020

cp's OEIS Frontend

A192761 Coefficient of x in the reduction by x^2->x+1 of the polynomial p(n,x) defined below in Comments.

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A192808 Constant term in the reduction of the polynomial (x^2 + 2)^n by x^3 -> x^2 + 2. See Comments.

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A192960 Constant term of the reduction by x^2 -> x+1 of the polynomial p(n,x) defined at Comments.

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A192967 Constant term of the reduction by x^2 -> x+1 of the polynomial p(n,x) defined at Comments.

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A192971 Constant term of the reduction by x^2 -> x+1 of the polynomial p(n,x) defined at Comments.

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A192617 Coefficient of x in the reduction of the n-th Fibonacci polynomial by x^3->x^2+x+1.

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A192651 Coefficient of x^2 in the reduction of the n-th Fibonacci polynomial by x^3->x^2+x+1. See Comments.