A192744 -id:A192744 - cp's OEIS frontend

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

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

A192974 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, 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)

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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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A192968 Coefficient of x in the reduction by x^2 -> x+1 of the polynomial p(n,x) defined at Comments.

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A192970 Coefficient of x in the reduction by x^2 -> x+1 of the polynomial p(n,x) defined at Comments.

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A192972 Coefficient of x in the reduction by x^2 -> x+1 of the polynomial p(n,x) defined at Comments.

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A192973 Constant term of the reduction by x^2 -> x+1 of the polynomial p(n,x) defined at Comments.

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