A192951 -id:A192951 - cp's OEIS frontend

A192969 Constant term of the reduction by x^2 -> x+1 of the polynomial p(n,x) defined at Comments.

1, 2, 6, 12, 23, 41, 71, 120, 200, 330, 541, 883, 1437, 2334, 3786, 6136, 9939, 16093, 26051, 42164, 68236, 110422, 178681, 289127, 467833, 756986, 1224846, 1981860, 3206735, 5188625, 8395391, 13584048, 21979472, 35563554, 57543061, 93106651

Offset: 0

Clark Kimberling, Jul 13 2011

nonn

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.

Daniel Suteu, Table of n, a(n) for n = 0..999
Index entries for linear recurrences with constant coefficients, signature (3,-2,-1,1).

Cf. A000045, A192232, A192744, A192951, A192970.

F:=Fibonacci;; List([0..40], n-> 2*F(n+2)+3*F(n+1)-n-4); # G. C. Greubel, Jul 11 2019

F:=Fibonacci; [2*F(n+2)+3*F(n+1)-n-4: n in [0..40]]; // G. C. Greubel, Jul 11 2019

F:= gfun:-rectoproc({a(0) = 1, a(1) = 2, a(n) = 1 + n + a(n-1) + a(n-2)},a(n),remember):
map(F, [$0..100]); # Robert Israel, Jan 18 2016

(* 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 *)
(* Second program *)
Table[2*Fibonacci[n+2]+3*Fibonacci[n+1]-n-4, {n,0,40}] (* G. C. Greubel, Jul 11 2019 *)

vector(40, n, n--; f=fibonacci; 2*f(n+2)+3*f(n+1)-n-4) \\ G. C. Greubel, Jul 11 2019

f=fibonacci; [2*f(n+2)+3*f(n+1)-n-4 for n in (0..40)] # G. C. Greubel, Jul 11 2019

func a((0)) { 1 }
func a((1)) { 2 }
func a(n) is cached { 1 + n + a(n-1) + a(n-2) }
100.times { |i| say a(i-1) }
# Daniel Suteu, Jan 12 2016

a(n) = 3*a(n-1) - 2*a(n-2) - a(n-3) + a(n-4).
G.f.: (1 - x + 2*x^2 - x^3)/((1-x-x^2)*(1-x)^2). - R. J. Mathar, May 11 2014
a(0) = 1; a(1) = 2; a(n) = 1 + n + a(n-1) + a(n-2). - Daniel Suteu, Jan 12 2016
a(n) = 2*Fibonacci(n+2) + 3*Fibonacci(n+1) - n - 4. - G. C. Greubel, Jul 11 2019

A192960 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, 7, 15, 29, 53, 93, 159, 267, 443, 729, 1193, 1945, 3163, 5135, 8327, 13493, 21853, 35381, 57271, 92691, 150003, 242737, 392785, 635569, 1028403, 1664023, 2692479, 4356557, 7049093, 11405709, 18454863, 29860635, 48315563, 78176265

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 (3,-2,-1,1).

Cf. A000045, A192232, A192744, A192951, A192961.

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

F:=Fibonacci; [2*F(n+4)-(2*n+5): 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+4]-(2*n+5), {n,0,40}]] (* G. C. Greubel, Jul 12 2019 *)

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

f=fibonacci; [2*f(n+4)-(2*n+5) 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.: (1+x)*(1-x+x^2)/((1-x-x^2)*(1-x)^2).
a(n) - a(n-1) = A019274(n+2). (End)
a(n) = 2*Fibonacci(n+4) - (2*n + 5). - G. C. Greubel, Jul 12 2019

A192967 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, 4, 9, 17, 31, 54, 92, 154, 255, 419, 685, 1116, 1814, 2944, 4773, 7733, 12523, 20274, 32816, 53110, 85947, 139079, 225049, 364152, 589226, 953404, 1542657, 2496089, 4038775, 6534894, 10573700, 17108626, 27682359, 44791019, 72473413, 117264468

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..1000
Index entries for linear recurrences with constant coefficients, signature (3,-2,-1,1).

Cf. A000045, A192232, A192744, A192951.

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

I:=[1, 0, 2, 4]; [n le 4 select I[n] else 3*Self(n-1)-2*Self(n-2)-Self(n-3)+Self(n-4): n in [1..40]]; // Vincenzo Librandi, Feb 20 2012

[3*Fibonacci(n+1) -n-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}] (* A192967 *)
u2 = Table[Coefficient[Part[t, n], x, 1], {n, 1, z}] (* A192968 *)
LinearRecurrence[{3,-2,-1,1}, {1,0,2,4}, 41] (* Vladimir Joseph Stephan Orlovsky, Feb 15 2012 *)
Table[3*Fibonacci[n+1] -n-2, {n,0,40}] (* G. C. Greubel, Jul 11 2019 *)

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

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

a(0)=1, a(1)=0, for n > 1, a(n) = a(n-1) + a(n-2) + n - 1. - Alex Ratushnyak, May 10 2012
a(n) = 3*a(n-1) - 2*a(n-2) - a(n-3) + a(n-4).
G.f.: (1 -3*x +4*x^2 -x^3)/((1-x-x^2)*(1-x)^2). - R. J. Mathar, May 11 2014
a(n) = 3*Fibonacci(n+1) - n - 2. - G. C. Greubel, Jul 11 2019

A192971 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, 9, 21, 44, 83, 149, 258, 437, 729, 1204, 1975, 3225, 5250, 8529, 13837, 22428, 36331, 58829, 95234, 154141, 249457, 403684, 653231, 1057009, 1710338, 2767449, 4477893, 7245452, 11723459, 18969029, 30692610, 49661765, 80354505

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 (3,-2,-1,1).

Cf. A000032, A000045, A192232, A192744, A192951, A192972.

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

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

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

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

A192953 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, 13, 26, 48, 85, 146, 246, 409, 674, 1104, 1801, 2930, 4758, 7717, 12506, 20256, 32797, 53090, 85926, 139057, 225026, 364128, 589201, 953378, 1542630, 2496061, 4038746, 6534864, 10573669, 17108594, 27682326, 44790985, 72473378

Offset: 0

Clark Kimberling, Jul 13 2011

The titular polynomials are defined recursively: p(n,x) = x*p(n-1,x) + 2n - 1, 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-> 3*F(n+2)-(2*n+3)); # G. C. Greubel, Jul 12 2019

F:=Fibonacci; [3*F(n+2)-(2*n+3): 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] + 2n - 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}] (* A111314 *)
u2 = Table[Coefficient[Part[t, n], x, 1], {n, 1, z}] (* A192953 *)
(* Second program *)
With[{F=Fibonacci}, Table[3*F[n+2]-(2*n+3), {n,0,40}]] (* G. C. Greubel, Jul 12 2019 *)

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

f=fibonacci; [3*f(n+2)-(2*n+3) 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).
G.f.: x*(1 -x +2*x^2)/((1-x-x^2)*(1-x)^2). - R. J. Mathar, Aug 01 2011
a(n) = -2*n - 3 + 3*A000045(n+2). - R. J. Mathar, Aug 01 2011
a(n) = A131300(n) - 1. - R. J. Mathar, Mar 24 2018
a(n) = 3*Fibonacci(n+2) - (2*n+3). - G. C. Greubel, Jul 12 2019

A192954 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, 5, 11, 23, 43, 77, 133, 225, 375, 619, 1015, 1657, 2697, 4381, 7107, 11519, 18659, 30213, 48909, 79161, 128111, 207315, 335471, 542833, 878353, 1421237, 2299643, 3720935, 6020635, 9741629, 15762325, 25504017, 41266407, 66770491

Offset: 0

Clark Kimberling, Jul 13 2011

nonn

The titular polynomials are defined recursively: p(n,x) = x*p(n-1,x) + 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, A192955.

List([0..40], n-> 2*Lucas(1,-1,n+2)[2]-(2*n+5)); # G. C. Greubel, Jul 12 2019

[2*Lucas(n+2)-(2*n+5): 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;
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}] (* A192954 *)
u2 = Table[Coefficient[Part[t, n], x, 1], {n, 1, z}] (* A192955 *)
(* Second program *)
Table[2*LucasL[n+2]-(2*n+5), {n,0,40}] (* G. C. Greubel, Jul 12 2019 *)
LinearRecurrence[{3,-2,-1,1},{1,1,5,11},40] (* Harvey P. Dale, Jan 13 2022 *)

vector(40, n, n--; f=fibonacci; 2*(f(n+3)+f(n+1))-(2*n+5)) \\ G. C. Greubel, Jul 12 2019

[2*lucas_number2(n+2,1,-1)-(2*n+5) 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.: (1 -2*x +4*x^2 -x^3)/((1-x-x^2)*(1-x)^2).
a(n) - a(n-1) = A168674(n-1). (End)
a(n) = 2*Lucas(n+2) - (2*n+5). - G. C. Greubel, Jul 12 2019

A192955 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, 7, 18, 41, 84, 161, 294, 519, 894, 1513, 2528, 4185, 6882, 11263, 18370, 29889, 48548, 78761, 127670, 206831, 334942, 542257, 877728, 1420561, 2298914, 3720151, 6019794, 9740729, 15761364, 25502993, 41265318, 66769335, 108035742

Offset: 0

Clark Kimberling, Jul 13 2011

nonn

The titular polynomials are defined recursively: p(n,x) = x*p(n-1,x) + 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, A192955.

List([0..40], n-> 2*Lucas(1,-1,n+3)[2]-(n^2+4*n+8)); # G. C. Greubel, Jul 12 2019

[2*Lucas(n+3)-(n^2+4*n+8): 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;
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}] (* A192954 *)
u2 = Table[Coefficient[Part[t, n], x, 1], {n, 1, z}] (* A192955 *)
(* Second program *)Table[2*LucasL[n+3]-(n^2+4*n+8), {n,0,40}] (* G. C. Greubel, Jul 12 2019 *)

vector(40, n, n--; f=fibonacci; 2*(f(n+4)+f(n+2))-(n^2+4*n+8)) \\ G. C. Greubel, Jul 12 2019

[2*lucas_number2(n+3,1,-1)-(n^2+4*n+8) 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 08 2014: (Start)
G.f.: x*(1 -2*x +4*x^2 -x^3)/((1-x-x^2)*(1-x)^3).
a(n) - a(n-1) = A192954(n-1). (End)
a(n) = 2*Lucas(n+3) - (n^2+4*n+8). - G. C. Greubel, Jul 12 2019

A192956 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, 4, 9, 20, 38, 69, 120, 204, 341, 564, 926, 1513, 2464, 4004, 6497, 10532, 17062, 27629, 44728, 72396, 117165, 189604, 306814, 496465, 803328, 1299844, 2103225, 3403124, 5506406, 8909589, 14416056, 23325708, 37741829, 61067604, 98809502

Offset: 0

Clark Kimberling, Jul 13 2011

nonn

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.

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, A192957.

F:=Fibonacci;; List([0..40], n-> F(n+3)+4*F(n+1)-(2*n+5)); # G. C. Greubel, Jul 12 2019

F:=Fibonacci; [F(n+3)+4*F(n+1)-(2*n+5): 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 - 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}] (* A192956 *)
u2 = Table[Coefficient[Part[t, n], x, 1], {n, 1, z}] (* A192957 *)
(* Second program *)
With[{F=Fibonacci}, Table[F[n+3]+4*F[n+1]-(2*n+5), {n,0,40}]] (* G. C. Greubel, Jul 12 2019 *)

vector(40, n, n--; f=fibonacci; f(n+3)+4*f(n+1)-(2*n+5)) \\ G. C. Greubel, Jul 12 2019

f=fibonacci; [f(n+3)+4*f(n+1)-(2*n+5) 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.: (1 -3*x +6*x^2 -2*x^3)/((1-x-x^2)*(1-x)^2).
a(n) -2*a(n+1) +a(n+2) = A022096(n-3). (End)
a(n) = Fibonacci(n+3) + 4*Fibonacci(n+1) - (2*n+5). - G. C. Greubel, Jul 12 2019

A192957 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, 5, 14, 34, 72, 141, 261, 465, 806, 1370, 2296, 3809, 6273, 10277, 16774, 27306, 44368, 71997, 116725, 189121, 306286, 495890, 802704, 1299169, 2102497, 3402341, 5505566, 8908690, 14415096, 23324685, 37740741, 61066449, 98808278

Offset: 0

Clark Kimberling, Jul 13 2011

nonn

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.

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, A192956.

F:=Fibonacci;; List([0..40], n-> F(n+4)+4*F(n+2)-(n^2+4*n+7)); # G. C. Greubel, Jul 12 2019

F:=Fibonacci; [F(n+4)+4*F(n+2)-(n^2+4*n+7): 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 - 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}] (* A192956 *)
u2 = Table[Coefficient[Part[t, n], x, 1], {n, 1, z}] (* A192957 *)
(* Second program *)With[{F=Fibonacci}, Table[F[n+4]+4*F[n+2]-(n^2+4*n+7), {n,0,40}]] (* G. C. Greubel, Jul 12 2019 *)

vector(40, n, n--; f=fibonacci; f(n+4)+4*f(n+2)-(n^2+4*n+7)) \\ G. C. Greubel, Jul 12 2019

f=fibonacci; [f(n+4)+4*f(n+2)-(n^2+4*n+7) 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 -3*x +6*x^2 -2*x^3)/((1-x-x^2)*(1-x)^3).
a(n) - a(n-1) = A192956(n-1). (End)
a(n) = Fibonacci(n+4) + 4*Fibonacci(n+2) - (n^2+4*n+7). - G. C. Greubel, Jul 12 2019

A192958 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, 3, 7, 17, 33, 61, 107, 183, 307, 509, 837, 1369, 2231, 3627, 5887, 9545, 15465, 25045, 40547, 65631, 106219, 171893, 278157, 450097, 728303, 1178451, 1906807, 3085313, 4992177, 8077549, 13069787, 21147399, 34217251, 55364717, 89582037

Offset: 0

Clark Kimberling, Jul 13 2011

sign

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 (3,-2,-1,1).

Cf. A000045, A192232, A192744, A192951, A192959.

F:=Fibonacci;; List([0..40], n-> 6*F(n+1)-(2*n+5)); # G. C. Greubel, Jul 12 2019

F:=Fibonacci; [6*F(n+1)-(2*n+5): 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+1]-(2*n+5), {n,0,40}]] (* G. C. Greubel, Jul 12 2019 *)

vector(40, n, n--; f=fibonacci; 6*f(n+1)-(2*n+5)) \\ G. C. Greubel, Jul 12 2019

f=fibonacci; [6*f(n+1)-(2*n+5) 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.: (1 -4*x +8*x^2 -3*x^3)/((1-x-x^2)*(1-x)^2).
a(n) - 2*a(n-1) +a(n-2) = A022089(n-3). (End)
a(n) = 6*Fibonacci(n+1) - (2*n+5). - G. C. Greubel, Jul 12 2019

cp's OEIS Frontend

A192969 Constant term of the reduction by x^2 -> x+1 of the polynomial p(n,x) defined at Comments.

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A192960 Constant term of the reduction by x^2 -> x+1 of the polynomial p(n,x) defined at Comments.

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A192967 Constant term of the reduction by x^2 -> x+1 of the polynomial p(n,x) defined at Comments.

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A192971 Constant term of the reduction by x^2 -> x+1 of the polynomial p(n,x) defined at Comments.

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A192953 Coefficient of x in the reduction by x^2 -> x+1 of the polynomial p(n,x) defined at Comments.

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A192954 Constant term of the reduction by x^2 -> x+1 of the polynomial p(n,x) defined at Comments.

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