A192951 -id:A192951 - cp's OEIS frontend

A192974 Coefficient of x in the reduction by x^2 -> x+1 of the polynomial p(n,x) defined at Comments.

0, 1, 4, 14, 37, 84, 172, 329, 600, 1058, 1821, 3080, 5144, 8513, 13996, 22902, 37349, 60764, 98692, 160105, 259520, 420426, 680829, 1102224, 1784112, 2887489, 4672852, 7561694, 12236005, 19799268, 32036956, 51838025, 83876904, 135716978

Offset: 0

Clark Kimberling, Jul 13 2011

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

Cf. A000032, A000045, A192232, A192744, A192951, A192971.

F:=Fibonacci;; List([0..40], n-> F(n+6)+3*F(n+4) -(2*n^2+8*n+17)); # G. C. Greubel, Jul 24 2019

[Fibonacci(n+7)+Lucas(n+3)-2*n*(n+4)-17: n in [0..40]]; // Vincenzo Librandi, Jul 15 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 +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}] (* A192973 *)
u2 = Table[Coefficient[Part[t, n], x, 1], {n, 1, z}] (* A192974 *)
(* Additional programs *)
Table[Fibonacci[n+7] +LucasL[n+3] -2n(n+4) -17, {n,0,40}] (* Vincenzo Librandi, Jul 15 2019 *)

a(n)=fibonacci(n+7) + fibonacci(2*n+6)/fibonacci(n+3) - 2*n*(n+4) - 17 \\ Richard N. Smith, Jul 14 2019

f=fibonacci; [f(n+6)+3*f(n+4) -(2*n^2+8*n+17) 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+3*x^2)/((1-x-x^2)*(1-x)^3). - R. J. Mathar, May 11 2014
a(n) = Fibonacci(n+7) + Lucas(n+3) - 2*n*(n+4) - 17. - Ehren Metcalfe, Jul 14 2019

A192975 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, 8, 19, 41, 78, 141, 245, 416, 695, 1149, 1886, 3081, 5017, 8152, 13227, 21441, 34734, 56245, 91053, 147376, 238511, 385973, 624574, 1010641, 1635313, 2646056, 4281475, 6927641, 11209230, 18136989, 29346341, 47483456, 76829927

Offset: 0

Clark Kimberling, Jul 13 2011

nonn

The titular polynomials are defined recursively: p(n,x) = x*p(n-1,x) - 1 + 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, A192976.

F:=Fibonacci;; List([0..40], n-> 4*F(n+3)+3*F(n+1) -2*(2*n+5)); # G. C. Greubel, Jul 24 2019

[Fibonacci(n+3)+3*Lucas(n+2)-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 -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}] (* A192975 *)
u2 = Table[Coefficient[Part[t, n], x, 1], {n, 1, z}] (* A192976 *)
(* Additional programs *)
Table[Fibonacci[n+3]+3*LucasL[n+2] -2*(2*n+5), {n,0,40}] (* G. C. Greubel, Jul 24 2019 *)

vector(40, n, n--; f=fibonacci; 4*f(n+3)+3*f(n+1) -2*(2*n+5)) \\ G. C. Greubel, Jul 24 2019

f=fibonacci; [4*f(n+3)+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.: (1-2*x+7*x^2-2*x^3)/((1-x-x^2)*(1-x)^2). - R. J. Mathar, May 11 2014
a(n) = Fibonacci(n+3) + 3*Lucas(n+2) - 2*(2*n+5). - G. C. Greubel, Jul 24 2019

A192976 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, 10, 29, 70, 148, 289, 534, 950, 1645, 2794, 4680, 7761, 12778, 20930, 34157, 55598, 90332, 146577, 237630, 385006, 623517, 1009490, 1634064, 2644705, 4280018, 6926074, 11207549, 18135190, 29344420, 47481409, 76827750, 124311206

Offset: 0

Clark Kimberling, Jul 13 2011

nonn

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

Cf. A000032, A000045, A192232, A192744, A192951, A192975.

F:=Fibonacci;; List([0..40], n-> 4*F(n+4)+3*F(n+2)-(2*n^2+8*n+15)); # G. C. Greubel, Jul 24 2019

[Fibonacci(n+4)+3*Lucas(n+3)-(2*n^2+8*n+15): 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 -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}] (* A192975 *)
u2 = Table[Coefficient[Part[t, n], x, 1], {n, 1, z}] (* A192976 *)
(* Additional programs *)
Table[Fibonacci[n+4]+3*LucasL[n+3] -(2*n^2+8*n+15), {n,0,40}] (* G. C. Greubel, Jul 24 2019 *)

vector(40, n, n--; f=fibonacci; 4*f(n+4)+3*f(n+2) -(2*n^2 + 8*n + 15)) \\ G. C. Greubel, Jul 24 2019

f=fibonacci; [4*f(n+4)+3*f(n+2) -(2*n^2+8*n+15) 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+7*x^2-2*x^3)/((1-x-x^2)*(1-x)^3). - R. J. Mathar, May 11 2014
a(n) = Fibonacci(n+4) + 3*Lucas(n+3) - (2*n^2 + 8*n + 15). - G. C. Greubel, Jul 24 2019

A192978 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, 4, 12, 29, 62, 122, 227, 406, 706, 1203, 2020, 3356, 5533, 9072, 14816, 24129, 39218, 63654, 103215, 167250, 270886, 438599, 709992, 1149144, 1859737, 3009532, 4869972, 7880261, 12751046, 20632178, 33384155, 54017326, 87402538

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.

Define a triangle by T(n,0) = n*(n+1) + 1, T(n,n)=1, and T(r,c) = T(r-1,c-1) + T(r-2,c-1). The sum of the terms in row(n) is a(n+1). - J. M. Bergot, Apr 14 2013

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.

List([0..40], n-> Lucas(1,-1, n+5)[2] -(n^2+5*n+11)); # G. C. Greubel, Jul 24 2019

[Lucas(n+5)-(n^2+5*n+11): 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}] (* A027181 *)
u2 = Table[Coefficient[Part[t, n], x, 1], {n, 1, z}] (* A192978 *)
(* Additional programs *)
CoefficientList[Series[x*(1+x^2)/((1-x-x^2)*(1-x)^3), {x, 0, 40}], x] (* Vincenzo Librandi, May 13 2014 *)
Table[LucasL[n+5] -(n^2+5*n+11), {n,0,40}] (* G. C. Greubel, Jul 24 2019 *)
LinearRecurrence[{4,-5,1,2,-1},{0,1,4,12,29},40] (* Harvey P. Dale, Dec 24 2023 *)

vector(40, n, n--; f=fibonacci; f(n+6)+f(n+4) -(n^2+5*n+11)) \\ G. C. Greubel, Jul 24 2019

[lucas_number2(n+5, 1,-1) -(n^2+5*n+11) 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)/((1-x-x^2)*(1-x)^3). - R. J. Mathar, May 12 2014
a(n) = Lucas(n+5) - n*(n+5) - 11. - Ehren Metcalfe, Jul 13 2019
From Stefano Spezia, Jul 13 2019: (Start)
a(n) = (1/2)*(-22 + (11 - 5*sqrt(5))*((1/2)*(1 - sqrt(5)))^n + 11*((1/2)* (1 + sqrt(5)))^n + 5*sqrt(5)*((1/2)*(1 + sqrt(5)))^n - 10*n - 2*n^2).
E.g.f.: (1/2)*(2 + sqrt(5))*((-47 + 21*sqrt(5))*exp(-(1/2)*(-1 + sqrt(5))*x) + (3 + sqrt(5))*exp((1/2)*(1 + sqrt(5))*x) - 2*(-2 + sqrt(5))*exp(x)*(11 + 6*x + x^2)).
(End)

A192979 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, 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

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A192974 Coefficient of x in the reduction by x^2 -> x+1 of the polynomial p(n,x) defined at Comments.

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A192975 Constant term of the reduction by x^2 -> x+1 of the polynomial p(n,x) defined at Comments.

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A192976 Coefficient of x in the reduction by x^2 -> x+1 of the polynomial p(n,x) defined at Comments.

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A192978 Coefficient of x in the reduction by x^2 -> x+1 of the polynomial p(n,x) defined at Comments.

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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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