A192744 -id:A192744 - cp's OEIS frontend

A192979 Constant term of the reduction by x^2 -> x+1 of the polynomial p(n,x) defined at Comments.

1, 1, 4, 9, 19, 36, 65, 113, 192, 321, 531, 872, 1425, 2321, 3772, 6121, 9923, 16076, 26033, 42145, 68216, 110401, 178659, 289104, 467809, 756961, 1224820, 1981833, 3206707, 5188596, 8395361, 13584017, 21979440, 35563521, 57543027, 93106616

Offset: 0

Clark Kimberling, Jul 13 2011

The titular polynomials are defined recursively: p(n,x) = x*p(n-1,x) + 1 - n + 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, A192980.

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

[Fibonacci(n+3)+Lucas(n+2)-2*(n+2): 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] +n^2-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}] (* A192979 *)
u2 = Table[Coefficient[Part[t, n], x, 1], {n, 1, z}] (* A192980 *)
(* Additional programs *)
Table[Fibonacci[n+3]+LucasL[n+2] -2*(n+2), {n,0,40}] (* G. C. Greubel, Jul 24 2019 *)

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

f=fibonacci; [2*f(n+3)+f(n+1) -2*(n+2) 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.: (1-2*x+3*x^2)/((1-x)^2*(1-x-x^2)). - Colin Barker, May 11 2014
a(n) = Fibonacci(n+3) + Lucas(n+2) - 2*(n+2). - G. C. Greubel, Jul 24 2019

A192980 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, 15, 34, 70, 135, 248, 440, 761, 1292, 2164, 3589, 5910, 9682, 15803, 25726, 41802, 67835, 109980, 178196, 288597, 467256, 756360, 1224169, 1981130, 3205950, 5187783, 8394490, 13583086, 21978447, 35562464, 57541904, 93105425

Offset: 0

Clark Kimberling, Jul 13 2011

nonn

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

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (4,-5,1,2,-1).

Cf. A000032, A000045, A192232, A192744, A192951, A192979.

F:=Fibonacci;; List([0..40], n-> 2*F(n+4)+F(n+2) -(n^2+3*n+7)); # G. C. Greubel, Jul 24 2019

[Fibonacci(n+4)+Lucas(n+3)-(n^2+3*n+7): 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] +n^2-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}] (* A192979 *)
u2 = Table[Coefficient[Part[t, n], x, 1], {n, 1, z}] (* A192980 *)
(* Additional programs *)
CoefficientList[Series[x*(1-2*x+3*x^2)/((1-x-x^2)*(1-x)^3), {x,0,40}], x] (* Vincenzo Librandi, May 13 2014 *)
Table[Fibonacci[n+4]+LucasL[n+3] -(n^2+3*n+7), {n,0,40}] (* G. C. Greubel, Jul 24 2019 *)

vector(40, n, n--; f=fibonacci; 2*f(n+4)+f(n+2) -(n^2+3*n+7)) \\ G. C. Greubel, Jul 24 2019

f=fibonacci; [2*f(n+4)+f(n+2) -(n^2+3*n+7) for n in (0..40)] # G. C. Greubel, Jul 24 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)/((1-x-x^2)*(1-x)^3). - R. J. Mathar, May 12 2014
a(n) = Fibonacci(n+4) + Lucas(n+3) - (n^2 + 3*n + 7). - G. C. Greubel, Jul 24 2019

A192981 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, 12, 24, 45, 80, 138, 233, 388, 640, 1049, 1712, 2786, 4525, 7340, 11896, 19269, 31200, 50506, 81745, 132292, 214080, 346417, 560544, 907010, 1467605, 2374668, 3842328, 6217053, 10059440, 16276554, 26336057, 42612676, 68948800

Offset: 0

Clark Kimberling, Jul 14 2011

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

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, A192982.

F:=Fibonacci;; List([0..40], n-> F(n+3)+2*F(n+1) -(2*n+3)); # G. C. Greubel, Jul 25 2019

F:=Fibonacci; [F(n+3)+2*F(n+1) -(2*n+3): n in [0..40]]; // G. C. Greubel, Jul 25 2019

(* First program *)
q = x^2; s = x + 1; z = 40;
p[0, x]:= 1;
p[n_, x_]:= x*p[n-1, x] + (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}] (* A192981 *)
u2 = Table[Coefficient[Part[t, n], x, 1], {n, 1, z}] (* A192982 *)
(* Additional programs *)
Table[LucasL[n+2]Fibonacci[n+1]-(2*n+3), {n,0,40}] (* _G. C. Greubel, Jul 25 2019 *)

vector(40, n, n--; f=fibonacci; f(n+3)+2*f(n+1) -(2*n+3)) \\ G. C. Greubel, Jul 25 2019

f=fibonacci; [f(n+3)+2*f(n+1) -(2*n+3) for n in (0..40)] # G. C. Greubel, Jul 25 2019

a(n) = 3*a(n-1) - 2*a(n-2) - a(n-3) + a(n-4).
G.f.: (1 - 3*x + 4*x^2)/((1 - x)^2*(1 - x - x^2)). - Colin Barker, May 11 2014
a(n) = Lucas(n+2) + Fibonacci(n+1) - (2*n+3). - G. C. Greubel, Jul 25 2019

A192982 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, 20, 44, 89, 169, 307, 540, 928, 1568, 2617, 4329, 7115, 11640, 18980, 30876, 50145, 81345, 131851, 213596, 345888, 559968, 906385, 1466929, 2373939, 3841544, 6216212, 10058540, 16275593, 26335033, 42611587, 68947644, 111560320

Offset: 0

Clark Kimberling, Jul 14 2011

The titular polynomials are defined recursively: p(n,x) = x*p(n-1,x) + (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 (4,-5,1,2,-1).

Cf. A000032, A000045, A192232, A192744, A192951, A192981.

F:=Fibonacci;; List([0..40], n-> F(n+4)+2*F(n+2) -(n^2+2*n+5)); # G. C. Greubel, Jul 25 2019

F:=Fibonacci; [F(n+4)+2*F(n+2) -(n^2+2*n+5): n in [0..40]]; // G. C. Greubel, Jul 25 2019

(* First program *)
q = x^2; s = x + 1; z = 40;
p[0, x]:= 1;
p[n_, x_]:= x*p[n-1, x] + (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}] (* A192981 *)
u2 = Table[Coefficient[Part[t, n], x, 1], {n, 1, z}] (* A192982 *)
(* Additional programs *)
CoefficientList[Series[x*(1-3*x+4*x^2)/((1-x-x^2)*(1-x)^3), {x, 0, 40}], x] (* Vincenzo Librandi, May 13 2014 *)
Table[LucasL[n+3]+Fibonacci[n+2]-(n^2+2*n+5), {n,0,40}] (* G. C. Greubel, Jul 25 2019 *)

vector(40, n, n--; f=fibonacci; f(n+4)+2*f(n+2) -(n^2+2*n+5)) \\ G. C. Greubel, Jul 25 2019

f=fibonacci; [f(n+4)+2*f(n+2) -(n^2+2*n+5) for n in (0..40)] # G. C. Greubel, Jul 25 2019

a(n) = 4*a(n-1) - 5*a(n-2) + a(n-3) + 2*a(n-4) - a(n-5).
G.f.: x*(1-3*x+4*x^2)/((1-x-x^2)*(1-x)^3). - R. J. Mathar, May 12 2014
a(n) = Lucas(n+3) + Fibonacci(n+2) - (n^2 + 2*n + 5). - G. C. Greubel, Jul 25 2019

A193046 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, 17, 83, 275, 727, 1673, 3505, 6873, 12843, 23155, 40639, 69889, 118353, 198097, 328659, 541667, 888311, 1451433, 2365089, 3846201, 6245771, 10131747, 16423103, 26606785, 43088737, 69761873, 112925075, 182770163, 295787863

Offset: 0

Clark Kimberling, Jul 15 2011

The titular polynomials are defined recursively: p(n,x)=x*p(n-1,x)+n^4, 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, A193047.

q = x^2; s = x + 1; z = 40;
p[0, x] := 1;
p[n_, x_] := x*p[n - 1, x] + n^4;
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}]   (* A193046 *)
u2 = Table[Coefficient[Part[t, n], x, 1], {n, 1, z}]   (* A193047 *)

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^5-6*x^4-x^3-21*x^2+4*x-1) / ((x-1)^4*(x^2+x-1)). - Colin Barker, May 11 2014

A192389 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, 3, 9, 22, 48, 96, 181, 327, 573, 982, 1656, 2760, 4561, 7491, 12249, 19966, 32472, 52728, 85525, 138615, 224541, 363598, 588624, 952752, 1541953, 2495331, 4037961, 6534022, 10572768, 17107632, 27681301, 44789895, 72472221, 117263206

Offset: 0

Clark Kimberling, Jul 13 2011

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

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (4,-5,1,2,-1).

Cf. A000045, A192232, A192744, A192951.

List([0..40], n-> 3*Fibonacci(n+2)-n*(n+4)-9); # G. C. Greubel, Jul 24 2019

[3*Fibonacci(n+2)-n*(n+4)-9: n in [0..40]]; // G. C. Greubel, Jul 24 2019

(* First program *)
p[0, x]:= 1; p[n_, x_]:= x*p[n-1, x] +n^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}] (* A192953 *)
u2 = Table[Coefficient[Part[t, n], x, 1], {n, 1, z}] (* A192389 *)
(* Additional programs *)
CoefficientList[Series[x*(1-x+2*x^2)/((1-x)^3*(1-x-x^2)), {x, 0, 40}], x] (* Vincenzo Librandi, May 13 2014 *)
Table[3*Fibonacci[n+4] -n*(n+4)-9, {n,0,40}] (* G. C. Greubel, Jul 24 2019 *)

Vec(x*(1-x+2*x^2)/((1-x)^3*(1-x-x^2)) + O(x^40)) \\ Colin Barker, May 12 2014

vector(40, n, n--; 3*fibonacci(n+2)-n*(n+4)-9) \\ G. C. Greubel, Jul 24 2019

[3*fibonacci(n+2)-n*(n+4)-9 for n in (0..40)] # G. C. Greubel, Jul 24 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-x+2*x^2)/((1-x)^3*(1-x-x^2)). - Colin Barker, May 12 2014
a(n) = 3*Fibonacci(n+4) - n*(n+4) - 9. - Ehren Metcalfe, Jul 13 2019

A192745 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, 2, 5, 13, 42, 175, 937, 6152, 47409, 416441, 4092650, 44425891, 527520141, 6798966832, 94504778173, 1408978113005, 22426272779178, 379522678988183, 6804322657495361, 128828945745315544, 2568535276579450905, 53788306394034206449

Offset: 0

Clark Kimberling, Jul 09 2011

nonn

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

			The first six polynomials and their reductions are shown here:
1 -> 1
1+x -> 1+x
2+x+x^2 -> 3+2x
6+2x+x^2+x^3 -> 8+5x
24+6x+2x^2+x^4+x^5 -> 29+13x
From those, read A192744=(1,1,3,8,29,...) and A192745=(0,1,2,5,13,...).

A192744, A192232.

Mathematica
```
(See A192744.)
```

G.f.: x/(1-x-x^2)/Q(0), where Q(k)= 1 - x*(k+1)/(1 - x*(k+1)/Q(k+1)); (continued fraction). - Sergei N. Gladkovskii, May 20 2013
Conjecture: a(n) -n*a(n-1) +(n-2)*a(n-2) +(n-1)*a(n-3)=0. - R. J. Mathar, May 04 2014
a(n) = Sum_{k=0..n} k!*Fibonacci(n-k). - Greg Dresden, Dec 03 2021
a(n) ~ (n-1)!. - Vaclav Kotesovec, Dec 03 2021

A192747 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, 6, 15, 32, 61, 110, 191, 324, 541, 894, 1467, 2396, 3901, 6338, 10283, 16668, 27001, 43722, 70779, 114560, 185401, 300026, 485495, 785592, 1271161, 2056830, 3328071, 5384984, 8713141, 14098214, 22811447, 36909756, 59721301, 96631158

Offset: 1

Clark Kimberling, Jul 09 2011

nonn

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

Cf. A192744, A192232, A192746.

Mathematica
```
(See A192746.)
```

Conjecture: G.f.: x^2*(-1-3*x+x^2) / ( (x^2+x-1)*(x-1)^2 ), so the first differences are in A192746. - R. J. Mathar, May 04 2014

A192748 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

0, 1, 4, 11, 24, 47, 86, 151, 258, 433, 718, 1181, 1932, 3149, 5120, 8311, 13476, 21835, 35362, 57251, 92670, 149981, 242714, 392761, 635544, 1028377, 1663996, 2692451, 4356528, 7049063, 11405678, 18454831, 29860602, 48315529, 78176230

Offset: 1

Clark Kimberling, Jul 09 2011

nonn

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

Cf. A192744, A192232.

q = x^2; s = x + 1; z = 40;
p[0, n_] := 1; p[n_, x_] := x*p[n - 1, x] + 3 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}]
  (* A154691 *)
u2 = Table[Coefficient[Part[t, n], x, 1], {n, 1, z}]
  (* A192748 *)

Conjecture: G.f.: -x^2*(1+x+x^2) / ( (x^2+x-1)*(x-1)^2 ), so the first differences are in A154691. - R. J. Mathar, May 04 2014

A192749 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

0, 1, 6, 16, 35, 68, 124, 217, 370, 620, 1027, 1688, 2760, 4497, 7310, 11864, 19235, 31164, 50468, 81705, 132250, 214036, 346371, 560496, 906960, 1467553, 2374614, 3842272, 6216995, 10059380, 16276492, 26335993, 42612610, 68948732, 111561475

Offset: 0

Clark Kimberling, Jul 09 2011

The titular polynomial is defined recursively by p(n,x)=x*(n-1,x)+4n+1 for n>0, where p(0,x)=1. For discussions of polynomial reduction, see A192232 and A192744.

a(n+1) is the row sum of row n of the triangle defined by T(n,1)=n*(n-1)+1, T(n,n)=2*n-1, n>=1, and T(r,c)=T(r-1,c)+T(r-2,c-1). The triangle starts 1; 3,3; 7,4,5; 13,7,8,7; 21,14,12,12,9; - J. M. Bergot, Apr 26 2013

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

Cf. A192744, A192232.

q = x^2; s = x + 1; z = 40;
p[0, n_] := 1; p[n_, x_] := x*p[n - 1, x] + 4 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}]
(* A053311 *)
u2 = Table[Coefficient[Part[t, n], x, 1], {n, 1, z}]
(* A192749 *)

G.f. -x*(1+3*x) / ( (x^2+x-1)*(x-1)^2 ). a(n+1)-a(n) = A053311(n). - R. J. Mathar, Apr 29 2013

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A192979 Constant term of the reduction by x^2 -> x+1 of the polynomial p(n,x) defined at Comments.

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A192980 Coefficient of x in the reduction by x^2 -> x+1 of the polynomial p(n,x) defined at Comments.

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A192981 Constant term of the reduction by x^2 -> x+1 of the polynomial p(n,x) defined at Comments.

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A192982 Coefficient of x in the reduction by x^2 -> x+1 of the polynomial p(n,x) defined at Comments.

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A193046 Constant term of the reduction by x^2->x+1 of the polynomial p(n,x) defined at Comments.

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A192389 Coefficient of x in the reduction by x^2 -> x+1 of the polynomial p(n,x) defined at Comments.

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