A192951 -id:A192951 - cp's OEIS frontend

A192952 Coefficient of x in the reduction by x^2 -> x+1 of the polynomial p(n,x) defined at Comments.

0, 1, 2, 7, 16, 33, 62, 111, 192, 325, 542, 895, 1468, 2397, 3902, 6339, 10284, 16669, 27002, 43723, 70780, 114561, 185402, 300027, 485496, 785593, 1271162, 2056831, 3328072, 5384985, 8713142, 14098215, 22811448, 36909757, 59721302, 96631159

Offset: 0

Clark Kimberling, Jul 13 2011

nonn

The titular polynomials are defined recursively: p(n,x) = x*p(n-1,x) + 3n - 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.

F:=Fibonacci;; List([0..40], n-> 4*F(n+2)-(3*n+4)); # G. C. Greubel, Jul 12 2019

F:=Fibonacci; [4*F(n+2)-(3*n+4): 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] + 3n - 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}] (* A192746 *)
u2 = Table[Coefficient[Part[t, n], x, 1], {n, 1, z}] (* A192952 *)
(* Second program *)
With[{F=Fibonacci}, Table[4*F[n+2]-(3*n+4), {n,0,40}]] (* G. C. Greubel, Jul 12 2019 *)

vector(40, n, n--; f=fibonacci; 4*f(n+2)-(3*n+4)) \\ G. C. Greubel, Jul 12 2019

f=fibonacci; [4*f(n+2)-(3*n+4) 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 08 2014: (Start)
G.f.: x*(1 -x +3*x^2)/((1-x-x^2)*(1-x)^2).
a(n) - a(n-1) = A192746(n-2). (End)
a(n) = 4*Fibonacci(n+2) - (3*n+4). - G. C. Greubel, Jul 12 2019

A192966 Coefficient of x in the reduction by x^2 -> x+1 of the polynomial p(n,x) defined at Comments.

Original entry on oeis.org

0, 1, 2, 6, 14, 30, 59, 110, 197, 343, 585, 983, 1634, 2695, 4420, 7220, 11760, 19116, 31029, 50316, 81535, 132061, 213827, 346141, 560244, 906685, 1467254, 2374290, 3841922, 6216618, 10058975, 16276058, 26335529, 42612115, 68948205, 111560915

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..400
Index entries for linear recurrences with constant coefficients, signature (4,-5,1,2,-1).

Cf. A000045, A192232, A192744, A192951.

F:=Fibonacci;; List([0..40], n-> F(n+4) +2*F(n+2) -(n^2+5*n+10)/2); # G. C. Greubel, Jul 11 2019

I:=[0, 1, 2, 6, 14]; [n le 5 select I[n] else 4*Self(n-1)-5*Self(n-2)+Self(n-3)+2*Self(n-4)-Self(n-5): n in [1..40]]; // Vincenzo Librandi, Nov 16 2011

F:=Fibonacci; [F(n+4) +2*F(n+2) -(n^2+5*n+10)/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}] (* A030119 *)
u2 = Table[Coefficient[Part[t, n], x, 1], {n, 1, z}] (* A192966 *)
LinearRecurrence[{4,-5,1,2,-1},{0,1,2,6,14},40] (* Vincenzo Librandi, Nov 16 2011 *)
Table[Fibonacci[n+4] +2*Fibonacci[n+2] -(n^2+5*n+10)/2, {n,0,40}] (* G. C. Greubel, Jul 11 2019 *)

vector(40, n, n--; f=fibonacci; f(n+4)+2*f(n+2)-(n^2+5*n+10)/2) \\ G. C. Greubel, Jul 11 2019

f=fibonacci; [f(n+4) +2*f(n+2) -(n^2+5*n+10)/2 for n in (0..40)] # G. C. Greubel, Jul 11 2019

a(n) = 4*a(n-1) - 5*a(n-2) + a(n-3) + 2*a(n-4) - a(n-5).
G.f.: x*(1 - 2*x + 3*x^2 - x^3)/((1-x-x^2)*(1-x)^3). - R. J. Mathar, May 11 2014
a(n) = Fibonacci(n+4) + 2*Fibonacci(n+2) - (n^2 + 5*n + 10)/2. - G. C. Greubel, Jul 11 2019

A193004 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, 1, 9, 29, 75, 165, 331, 623, 1123, 1963, 3357, 5651, 9405, 15525, 25477, 41633, 67831, 110281, 179031, 290339, 470511, 762111, 1234009, 1997639, 3233305, 5232745, 8468001, 13702853, 22173123, 35878413, 58054147, 93935351, 151992475

Offset: 0

Clark Kimberling, Jul 14 2011

The titular polynomials are defined recursively: p(n,x)=x*p(n-1,x)+n^3, with p(0,x)=1. For an introduction to reductions of polynomials by substitutions such as x^2->x+1, see A192232 and A192744.

Index entries for linear recurrences with constant coefficients, signature (4,-5,1,2,-1).

Cf. A192232, A192744, A192951, A193005.

q = x^2; s = x + 1; z = 40;
p[0, x] := 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}] (* A193004 *)
u2 = Table[Coefficient[Part[t, n], x, 1], {n, 1, z}] (* A193005 *)

Vec((x^4-3*x^3+10*x^2-3*x+1)/((x-1)^3*(x^2+x-1)) + O(x^100)) \\ Colin Barker, May 12 2014

a(n) = 4*a(n-1)-5*a(n-2)+a(n-3)+2*a(n-4)-a(n-5).

G.f.: (x^4-3*x^3+10*x^2-3*x+1) / ((x-1)^3*(x^2+x-1)). - Colin Barker, May 12 2014

A193006 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, 8, 27, 72, 160, 323, 610, 1102, 1929, 3302, 5562, 9261, 15292, 25100, 41023, 66844, 108684, 176447, 286158, 463746, 751165, 1216298, 1968982, 3186937, 5157720, 8346608, 13506435, 21855312, 35364184, 57222107, 92589082, 149814166

Offset: 0

Clark Kimberling, Jul 14 2011

The titular polynomials are defined recursively: p(n,x)=x*p(n-1,x)-1+n^3, with p(0,x)=1. For an introduction to reductions of polynomials by substitutions such as x^2->x+1, see A192232 and A192744.

Index entries for linear recurrences with constant coefficients, signature (4,-5,1,2,-1).

Cf. A192232, A192744, A192951, A193007.

q = x^2; s = x + 1; z = 40;
p[0, x] := 1;
p[n_, x_] := x*p[n - 1, x] + n^3 - 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}]
  (* A193006 *)
u2 = Table[Coefficient[Part[t, n], x, 1], {n, 1, z}]
  (* A193007 *)

a(n) = 4*a(n-1)-5*a(n-2)+a(n-3)+2*a(n-4)-a(n-5).

G.f.: (2*x^4-6*x^3+13*x^2-4*x+1)/((x-1)^3*(x^2+x-1)). [Colin Barker, Nov 12 2012]

A193008 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, 10, 31, 78, 170, 339, 636, 1144, 1997, 3412, 5740, 9549, 15758, 25854, 42243, 68818, 111878, 181615, 294520, 477276, 773057, 1251720, 2026296, 3279673, 5307770, 8589394, 13899271, 22490934, 36392642, 58886187, 95281620, 154170784

Offset: 0

Clark Kimberling, Jul 14 2011

The titular polynomials are defined recursively: p(n,x)=x*p(n-1,x)+1+n^3, with p(0,x)=1. For an introduction to reductions of polynomials by substitutions such as x^2->x+1, see A192232 and A192744.

Index entries for linear recurrences with constant coefficients, signature (4,-5,1,2,-1).

Cf. A192232, A192744, A192951, A193009.

q = x^2; s = x + 1; z = 40;
p[0, x] := 1;
p[n_, x_] := x*p[n - 1, x] + n^3 + 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}] (* A193008 *)
u2 = Table[Coefficient[Part[t, n], x, 1], {n, 1, z}] (* A193009 *)

a(n) = 4*a(n-1)-5*a(n-2)+a(n-3)+2*a(n-4)-a(n-5).

G.f.: (7*x^2-2*x+1)/((x-1)^3*(x^2+x-1)). [Colin Barker, Nov 12 2012]

A193044 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, 5, 13, 28, 56, 105, 189, 330, 564, 949, 1579, 2606, 4276, 6987, 11383, 18506, 30042, 48719, 78951, 127880, 207062, 335195, 542533, 878028, 1420886, 2299265, 3720529, 6020200, 9741164, 15761829, 25503489, 41265846, 66769896

Offset: 0

Clark Kimberling, Jul 15 2011

The titular polynomials are defined recursively: p(n,x)=x*p(n-1,x)+n(-1+n^2)/6, with p(0,x)=1. For an introduction to reductions of polynomials by substitutions such as x^2->x+1, see A192232 and A192744.

Index entries for linear recurrences with constant coefficients, signature (4,-5,1,2,-1).

Cf. A192232, A192744, A192951, A193045, A179991 (first differences).

q = x^2; s = x + 1; z = 40;
p[0, x] := 1;
p[n_, x_] := x*p[n - 1, x] + n (n^2 - 1)/6;
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}]
  (* A193044 *)
u2 = Table[Coefficient[Part[t, n], x, 1], {n, 1, z}]
  (* A193045 *)

a(n)=4*a(n-1)-5*a(n-2)+a(n-3)+2*a(n-4)-a(n-5).

G.f.: ( 1+7*x^2-4*x^3+x^4-4*x ) / ( (x^2+x-1)*(x-1)^3 ). - R. J. Mathar, May 04 2014

A193048 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, 1, 2, 8, 25, 68, 163, 357, 730, 1417, 2642, 4774, 8417, 14556, 24793, 41729, 69582, 115187, 189614, 310786, 507715, 827356, 1345697, 2185703, 3546350, 5749603, 9316428, 15089782, 24433615, 39554862, 64024437, 103620219, 167691032

Offset: 0

Clark Kimberling, Jul 15 2011

The titular polynomials are defined recursively: p(n,x)=x*p(n-1,x)+n(4-5*n^2+n^4)/120, with p(0,x)=1. For an introduction to reductions of polynomials by substitutions such as x^2->x+1, see A192232 and A192744.

Index entries for linear recurrences with constant coefficients, signature (6,-14,15,-5,-4,4,-1).

Cf. A192232, A192744, A192951, A193049, A193046.

q = x^2; s = x + 1; z = 40;
p[0, x] := 1;
p[n_, x_] := x*p[n - 1, x] + n (-1 + n^2) (-4 + n^2)/120;
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}] (* A193048 *)
u2 = Table[Coefficient[Part[t, n], x, 1], {n, 1, z}] (* A193049 *)

Vec((x^2-x+1)*(x^4-5*x^3+9*x^2-5*x+1)/((x-1)^5*(x^2+x-1)) + O(x^100)) \\ Colin Barker, May 12 2014

a(n) = 6*a(n-1)-14*a(n-2)+15*a(n-3)-5*a(n-4)-4*a(n-5)+4*a(n-6)-a(n-7).

G.f.: (x^2-x+1)*(x^4-5*x^3+9*x^2-5*x+1) / ((x-1)^5*(x^2+x-1)). - Colin Barker, May 12 2014

A193005 Coefficient of x in the reduction by x^2->x+1 of the polynomial p(n,x) defined at Comments.

Original entry on oeis.org

0, 1, 2, 11, 40, 115, 280, 611, 1234, 2357, 4320, 7677, 13328, 22733, 38258, 63735, 105368, 173199, 283480, 462511, 752850, 1223361, 1985472, 3219481, 5217120, 8450425, 13683170, 22151171, 35854024, 58027147, 93905560, 151959707, 245895058, 397887533

Offset: 0

Clark Kimberling, Jul 14 2011

nonn

The titular polynomials are defined recursively: p(n,x)=x*p(n-1,x)+n^3, with p(0,x)=1. For an introduction to reductions of polynomials by substitutions such as x^2->x+1, see A192232 and A192744.

Index entries for linear recurrences with constant coefficients, signature (5,-9,6,1,-3,1).

Cf. A192232, A192744, A192951, A193004, A000045.

Mathematica
```
(See A193004.)
```

a(n) = 5*a(n-1)-9*a(n-2)+6*a(n-3)+a(n-4)-3*a(n-5)+a(n-6).
G.f.: -x*(1-3*x+10*x^2-3*x^3+x^4) / ( (x^2+x-1)*(x-1)^4 ). - R. J. Mathar, May 12 2014
a(n) = 10*F(n+4) + 4*F(n+5) - 50 - 24*n - 6*n^2 - n^3, where F = A000045. - Greg Dresden, Jan 01 2021

A193007 Coefficient of x in the reduction by x^2->x+1 of the polynomial p(n,x) defined at Comments.

Original entry on oeis.org

0, 1, 1, 9, 36, 108, 268, 591, 1201, 2303, 4232, 7534, 13096, 22357, 37649, 62749, 103772, 170616, 279300, 455747, 741905, 1205651, 1956816, 3173114, 5142096, 8329033, 13486753, 21833361, 35339796, 57195108, 92559292, 149781399, 242370481

Offset: 0

Clark Kimberling, Jul 14 2011

The titular polynomials are defined recursively: p(n,x)=x*p(n-1,x)-1+n^3, with p(0,x)=1. For an introduction to reductions of polynomials by substitutions such as x^2->x+1, see A192232 and A192744.

Index entries for linear recurrences with constant coefficients, signature (5,-9,6,1,-3,1).

Cf. A192232, A192744, A192951, A193006.

(See A193006.)
LinearRecurrence[{5,-9,6,1,-3,1},{0,1,1,9,36,108},40] (* Harvey P. Dale, Sep 13 2021 *)

a(n) = 5*a(n-1)-9*a(n-2)+6*a(n-3)+a(n-4)-3*a(n-5)+a(n-6).

G.f.: -x*(2*x^4-6*x^3+13*x^2-4*x+1)/((x-1)^4*(x^2+x-1)). [Colin Barker, Nov 12 2012]

A193045 Coefficient of x in the reduction by x^2->x+1 of the polynomial p(n,x) defined at Comments.

Original entry on oeis.org

0, 1, 1, 3, 8, 21, 49, 105, 210, 399, 729, 1293, 2242, 3821, 6427, 10703, 17690, 29073, 47579, 77621, 126340, 205291, 333171, 540233, 875428, 1417961, 2295989, 3716875, 6016140, 9736669, 15756869, 25498033, 41259862, 66763351, 108029197

Offset: 0

Clark Kimberling, Jul 15 2011

nonn

The titular polynomials are defined recursively: p(n,x)=x*p(n-1,x)+n(-1+n^2)/6, with p(0,x)=1. For an introduction to reductions of polynomials by substitutions such as x^2->x+1, see A192232 and A192744.

Index entries for linear recurrences with constant coefficients, signature (5,-9,6,1,-3,1)

Cf. A192232, A192744, A192951, A193044.

Mathematica
```
(See A193044.)
```

a(n)=5*a(n-1)-9*a(n-2)+6*a(n-3)+a(n-4)-3*a(n-5)+a(n-6).

G.f.: -x*(1-4*x+7*x^2-4*x^3+x^4) / ( (x^2+x-1)*(x-1)^4 ). - R. J. Mathar, May 12 2014

cp's OEIS Frontend

A192952 Coefficient of x in the reduction by x^2 -> x+1 of the polynomial p(n,x) defined at Comments.

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A192966 Coefficient of x in the reduction by x^2 -> x+1 of the polynomial p(n,x) defined at Comments.

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A193004 Constant term of the reduction by x^2->x+1 of the polynomial p(n,x) defined at Comments.

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A193006 Constant term of the reduction by x^2->x+1 of the polynomial p(n,x) defined at Comments.

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A193008 Constant term of the reduction by x^2->x+1 of the polynomial p(n,x) defined at Comments.

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A193044 Constant term of the reduction by x^2->x+1 of the polynomial p(n,x) defined at Comments.

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A193048 Constant term of the reduction by x^2->x+1 of the polynomial p(n,x) defined at Comments.

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