A192951 -id:A192951 - cp's OEIS frontend

A192959 Coefficient of x in the reduction by x^2 -> x+1 of the polynomial p(n,x) defined at Comments.

0, 1, 0, 3, 10, 27, 60, 121, 228, 411, 718, 1227, 2064, 3433, 5664, 9291, 15178, 24723, 40188, 65233, 105780, 171411, 277630, 449523, 727680, 1177777, 1906080, 3084531, 4991338, 8076651, 13068828, 21146377, 34216164, 55363563, 89580814

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 (4,-5,1,2,-1).

Cf. A000045, A192232, A192744, A192951, A192958.

F:=Fibonacci;; List([0..40], n-> 6*F(n+2)-(n^2+4*n+6)); # G. C. Greubel, Jul 12 2019

F:=Fibonacci; [6*F(n+2)-(n^2+4*n+6): 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}] (* A192958 *)
u2 = Table[Coefficient[Part[t, n], x, 1], {n, 1, z}] (* A192959 *)
(* Second program *)
With[{F=Fibonacci}, Table[6*F[n+2]-(n^2+4*n+6), {n,0,40}]] (* G. C. Greubel, Jul 12 2019 *)

vector(40, n, n--; f=fibonacci; 6*f(n+2)-(n^2+4*n+6)) \\ G. C. Greubel, Jul 12 2019

f=fibonacci; [6*f(n+2)-(n^2+4*n+6) for n in (0..40)] # G. C. Greubel, Jul 12 2019

a(n) = 4*a(n-1) - 5*a(n-2) + a(n-3) + 2*a(n-4) - a(n-5).
From R. J. Mathar, May 09 2014: (Start)
G.f.: x*(1 -4*x +8*x^2 -3*x^3)/((1-x-x^2)*(1-x)^3).
a(n) - a(n-1) = A192958(n-1). (End)
a(n) = 6*Fibonacci(n+2) - (n^2 + 4*n + 6). - G. C. Greubel, Jul 12 2019

A192961 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, 11, 26, 55, 108, 201, 360, 627, 1070, 1799, 2992, 4937, 8100, 13235, 21562, 35055, 56908, 92289, 149560, 242251, 392254, 634991, 1027776, 1663345, 2691748, 4355771, 7048250, 11404807, 18453900, 29859609, 48314472, 78175107, 126490670

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 (4,-5,1,2,-1).

Cf. A000045, A192232, A192744, A192951, A192960.

F:=Fibonacci;; List([0..40], n-> 2*F(n+5)-(n^2+4*n+10)); # G. C. Greubel, Jul 12 2019

F:=Fibonacci; [2*F(n+5)-(n^2+4*n+10): 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+5]-(n^2+4*n+10), {n,0,40}]] (* G. C. Greubel, Jul 12 2019 *)
LinearRecurrence[{4,-5,1,2,-1},{0,1,4,11,26},40] (* Harvey P. Dale, Dec 30 2024 *)

vector(40, n, n--; f=fibonacci; 2*f(n+5)-(n^2+4*n+10)) \\ G. C. Greubel, Jul 12 2019

f=fibonacci; [2*f(n+5)-(n^2+4*n+10) 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.: x*(1+x)*(1-x+x^2)/((1-x-x^2)*(1-x)^3).
a(n) - a(n-1)= A192960(n-1). (End)
a(n) = 2*Fibonacci(n+5) - (n^2 + 4*n + 10). - G. C. Greubel, Jul 12 2019

A192962 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, 7, 15, 30, 55, 97, 166, 279, 463, 762, 1247, 2033, 3306, 5367, 8703, 14102, 22839, 36977, 59854, 96871, 156767, 253682, 410495, 664225, 1074770, 1739047, 2813871, 4552974, 7366903, 11919937, 19286902, 31206903, 50493871, 81700842

Offset: 1

Clark Kimberling, Jul 13 2011

nonn

The titular polynomials are defined recursively: p(n,x) = x*p(n-1,x) + 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 = 1..1000
Index entries for linear recurrences with constant coefficients, signature (3,-2,-1,1).

Cf. A000045, A192232, A192744, A192951, A192963.

F:=Fibonacci;; List([1..40], n-> 3*F(n+1) +4*F(n) -2*(n+2)); # G. C. Greubel, Jul 12 2019

F:=Fibonacci; [3*F(n+1) +4*F(n) -2*(n+2): n in [1..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(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}] (* A192962 *)
u2 = Table[Coefficient[Part[t, n], x, 1], {n, 1, z}] (* A192963 *)
(* Additional programs *)
CoefficientList[Series[(1-x+3x^2-x^3)/((1-x-x^2)(1-x)^2), {x, 0, 40}], x] (* Vincenzo Librandi, May 09 2014 *)
With[{F=Fibonacci}, Table[3*F[n+1]+4*F[n] -2*(n+2), {n,1,40}]] (* G. C. Greubel, Jul 12 2019 *)

vector(40, n, f=fibonacci; 3*f(n+1)+4*f(n)-2*(n+2)) \\ G. C. Greubel, Jul 12 2019

f=fibonacci; [3*f(n+1) +4*f(n) -2*(n+2) for n in (1..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.: x*(1 -x +3*x^2 -x^3)/((1-x-x^2)*(1-x)^2).
a(n) -2*a(n-1) + a(n-2) = A022120(n-4). (End)
a(n) = 3*Fibonacci(n+1) + 4*Fibonacci(n) - 2*(n+2). - G. C. Greubel, Jul 12 2019

A192963 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, 10, 25, 55, 110, 207, 373, 652, 1115, 1877, 3124, 5157, 8463, 13830, 22533, 36635, 59474, 96451, 156305, 253176, 409943, 663625, 1074120, 1738345, 2813115, 4552162, 7366033, 11919007, 19285910, 31205847, 50492749, 81699652, 132193523, 213894365, 346089148

Offset: 0

Clark Kimberling, Jul 13 2011

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

Cf. A000045, A192232, A192744, A192951, A192962.

F:=Fibonacci;; List([0..40], n-> 3*F(n+4) +4*F(n+2) -(n^2+5*n+10)); # G. C. Greubel, Jul 12 2019

F:=Fibonacci; [3*F(n+4) +4*F(n+2) -(n^2+5*n+10): 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(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}] (* A192962 *)
u2 = Table[Coefficient[Part[t, n], x, 1], {n, 1, z}] (* A192963 *)
(* Second program *)
With[{F=Fibonacci}, Table[3*F[n+3]+4*F[n+2] -(n^2+5*n+10), {n,0,40}]] (* G. C. Greubel, Jul 11 2019 *)
LinearRecurrence[{4,-5,1,2,-1},{0,1,3,10,25},50] (* Harvey P. Dale, Apr 03 2023 *)

vector(40, n, n--; f=fibonacci; 3*f(n+4)+4*f(n+2)-(n^2+5*n+10)) \\ G. C. Greubel, Jul 12 2019

f=fibonacci; [3*f(n+4) +4*f(n+2) -(n^2+5*n+10) for n in (0..40)] # G. C. Greubel, Jul 12 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 +3*x^2 -x^3)/((1-x-x^2)*(1-x)^3). - R. J. Mathar, May 11 2014
a(n) = 3*Fibonacci(n+3) + 4*Fibonacci(n+2) - (n^2 + 5*n +10). - G. C. Greubel, Jul 12 2019
E.g.f.: 2*exp(x/2)*(25*cosh(sqrt(5)*x/2) + 12*sqrt(5)*sinh(sqrt(5)*x/2))/5 - exp(x)*(10 + 6*x + x^2). - Stefano Spezia, Aug 30 2025

A192964 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, 3, 7, 16, 31, 57, 100, 171, 287, 476, 783, 1281, 2088, 3395, 5511, 8936, 14479, 23449, 37964, 61451, 99455, 160948, 260447, 421441, 681936, 1103427, 1785415, 2888896, 4674367, 7563321, 12237748, 19801131, 32038943, 51840140, 83879151

Offset: 0

Clark Kimberling, Jul 13 2011

nonn

The titular polynomials are defined recursively: p(n,x) = x*p(n-1,x) - 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. A000045, A192232, A192744, A192951, A192965.

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

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

(* First program *)
q = x^2; s = x + 1; z = 40;
p[0, x]:= 1;
p[n_, x_]:= x*p[n-1, x] + n(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}] (* A192964 *)
u2 = Table[Coefficient[Part[t, n], x, 1], {n, 1, z}] (* A192965 *)
(* Second program *)
With[{F=Fibonacci}, Table[F[n+3]+3*F[n+1] -2*(n+2), {n,0,40}]] (* G. C. Greubel, Jul 11 2019 *)

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

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

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

A192965 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, 4, 11, 27, 58, 115, 215, 386, 673, 1149, 1932, 3213, 5301, 8696, 14207, 23143, 37622, 61071, 99035, 160486, 259941, 420889, 681336, 1102777, 1784713, 2888140, 4673555, 7562451, 12236818, 19800139, 32037887, 51839018, 83877961

Offset: 0

Clark Kimberling, Jul 13 2011

nonn

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

Cf. A000045, A192232, A192744, A192951, A192964.

F:=Fibonacci;; List([0..40], n-> F(n+4) +3*F(n+2) -(n^2+3*n+6)); # G. C. Greubel, Jul 11 2019

F:=Fibonacci; [F(n+4) +3*F(n+2) -(n^2+3*n+6): n in [0..40]]; // G. C. Greubel, Jul 11 2019

(* First program *)
q = x^2; s = x + 1; z = 40;
p[0, x]:= 1;
p[n_, x_]:= x*p[n-1, x] + n(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}] (* A192964 *)
u2 = Table[Coefficient[Part[t, n], x, 1], {n, 1, z}] (* A192965 *)
(* Second program *)
With[{F=Fibonacci}, Table[F[n+4]+3*F[n+2] -(n^2+3*n+6), {n,0,40}]] (* G. C. Greubel, Jul 11 2019 *)

vector(40, n, n--; f=fibonacci; f(n+4)+3*f(n+2)-(n^2+3*n+6)) \\ G. C. Greubel, Jul 11 2019

f=fibonacci; [f(n+4) +3*f(n+2) -(n^2+3*n+6) 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 -3*x +5*x^2 -x^3)/((1-x-x^2)*(1-x)^3). - R. J. Mathar, May 11 2014
a(n) = Fibonacci(n+4) + 3*Fibonacci(n+2) - (n^2 + 3*n + 6). - G. C. Greubel, Jul 11 2019

A192968 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, 7, 16, 33, 64, 118, 210, 364, 619, 1038, 1723, 2839, 4653, 7597, 12370, 20103, 32626, 52900, 85716, 138826, 224773, 363852, 588901, 953053, 1542279, 2495683, 4038340, 6534429, 10573204, 17108098, 27681798, 44790424, 72472783

Offset: 0

Clark Kimberling, Jul 13 2011

nonn

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.

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. A000045, A192232, A192744, A192951, A192967.

List([0..40], n-> 3*Fibonacci(n+2) -(n^2+3*n+6)/2); # G. C. Greubel, Jul 11 2019

[3*Fibonacci(n+2) -(n^2+3*n+6)/2: n in [0..40]]; // G. C. Greubel, Jul 11 2019

Table[3*Fibonacci[n+2] -(n^2+3*n+6)/2, {n, 0, 40}] (* G. C. Greubel, Jul 11 2019 *)

vector(40, n, n--; 3*fibonacci(n+2) -(n^2+3*n+6)/2) \\ G. C. Greubel, Jul 11 2019

[3*fibonacci(n+2) -(n^2+3*n+6)/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 -3*x +4*x^2 -x^3)/((1-x-x^2)*(1-x)^3). - R. J. Mathar, May 11 2014
a(n) = 3*Fibonacci(n+2) -(n^2+3*n+6)/2. - G. C. Greubel, Jul 11 2019

A192970 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, 21, 44, 85, 156, 276, 476, 806, 1347, 2230, 3667, 6001, 9787, 15923, 25862, 41955, 68006, 110170, 178406, 288828, 467509, 756636, 1224469, 1981455, 3206301, 5188161, 8394896, 13583521, 21978912, 35562960, 57542432, 93105986

Offset: 0

Clark Kimberling, Jul 13 2011

The titular polynomials are defined recursively: p(n,x) = x*p(n-1,x) + n(n+3)/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, A192970.

F:=Fibonacci;; List([0..40], n-> 2*F(n+4)+F(n+2)-(n^2+7*n+14)/2); # G. C. Greubel, Jul 24 2019

[Fibonacci(n+4)+Lucas(n+3)-(n^2+7*n+14)/2: n in [0..40]]; // Vincenzo Librandi, Jul 13 2019

(* First progream *)
q = x^2; s = x + 1; z = 40;
p[0, x]:= 1;
p[n_, x_]:= x*p[n-1, x] + n*(n+3)/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}] (* A192969 *)
u2 = Table[Coefficient[Part[t, n], x, 1], {n, 1, z}] (* A192970 *)
(* Additional programs *)
CoefficientList[Series[x*(1-x+2*x^2-x^3)/((1-x-x^2)*(1-x)^3), {x,0,40}], x] (* Vincenzo Librandi, Jul 13 2019 *)
Table[LucasL[n+3]+Fibonacci[n+4]-(n^2+7*n+14)/2, {n,0,40}] (* G. C. Greubel, Jul 24 2019 *)

vector(40, n, n--; f=fibonacci; 2*f(n+4)+f(n+2)-(n^2+7*n+14)/2) \\ G. C. Greubel, Jul 24 2019

f=fibonacci; [2*f(n+4)+f(n+2)-(n^2+7*n+14)/2 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-x^3)/((1-x-x^2)*(1-x)^3). - R. J. Mathar, May 11 2014
a(n) = Fibonacci(n+4) + Lucas(n+3) - (n^2 + 7*n + 14)/2. - Ehren Metcalfe, Jul 13 2019

A192972 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, 12, 33, 77, 160, 309, 567, 1004, 1733, 2937, 4912, 8137, 13387, 21916, 35753, 58181, 94512, 153341, 248575, 402716, 652173, 1055857, 1709088, 2766097, 4476435, 7243884, 11721777, 18967229, 30690688, 49659717, 80352327, 130014092

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 (4,-5,1,2,-1).

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

F:=Fibonacci;; List([0..40], n-> 5*F(n+4)+F(n+2) -2*(n^2+4*n+8)); # G. C. Greubel, Jul 24 2019

F:=Fibonacci; [5*F(n+4)+F(n+2) -2*(n^2+4*n+8): 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+4]+F[n+2] -2*(n^2+4*n+8), {n,0,40}]] (* G. C. Greubel, Jul 24 2019 *)

vector(40, n, n--; f=fibonacci; 5*f(n+4)+f(n+2) -2*(n^2+4*n+8)) \\ G. C. Greubel, Jul 24 2019

f=fibonacci; [5*f(n+4)+f(n+2) -2*(n^2+4*n+8) 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+5*x^2-x^3)/((1-x-x^2)*(1-x)^3). - R. J. Mathar, May 11 2014
a(n) = 4*Fibonacci(n+4) + Lucas(n+3) - 2*(n^2+4*n+8). - G. C. Greubel, Jul 24 2019

A192973 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, 10, 23, 47, 88, 157, 271, 458, 763, 1259, 2064, 3369, 5483, 8906, 14447, 23415, 37928, 61413, 99415, 160906, 260403, 421395, 681888, 1103377, 1785363, 2888842, 4674311, 7563263, 12237688, 19801069, 32038879, 51840074, 83879083

Offset: 1

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.

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

Cf. A000032, A000045, A192232, A192744, A192951, A192974.

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

[Lucas(n+4)-Fibonacci(n-1)-2*(2*n+3): n in [1..40]]; // Vincenzo Librandi, Jul 14 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 *)
LinearRecurrence[{3, -2, -1, 1}, {1, 3, 10, 23}, 50] (* Vincenzo Librandi, Jul 14 2019 *)
With[{F = Fibonacci}, Table[F[n+4]+3*F[n+2] -2*(2*n+3), {n,40}]] (* G. C. Greubel, Jul 24 2019 *)

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

f=fibonacci; [f(n+4)+3*f(n+2) -2*(2*n+3) for n in (1..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.: x*(1+3*x^2)/((1-x-x^2)*(1-x)^2). - R. J. Mathar, May 11 2014
a(n) = Lucas(n+4) - Fibonacci(n-1) - 2*(2*n+3). - Ehren Metcalfe, Jul 13 2019

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A192959 Coefficient of x in the reduction by x^2 -> x+1 of the polynomial p(n,x) defined at Comments.

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A192961 Coefficient of x in the reduction by x^2 -> x+1 of the polynomial p(n,x) defined at Comments.

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A192962 Constant term of the reduction by x^2 -> x+1 of the polynomial p(n,x) defined at Comments.

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A192963 Coefficient of x in the reduction by x^2 -> x+1 of the polynomial p(n,x) defined at Comments.

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A192964 Constant term of the reduction by x^2->x+1 of the polynomial p(n,x) defined at Comments.

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A192965 Coefficient of x in the reduction by x^2 -> x+1 of the polynomial p(n,x) defined at Comments.

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