A192744 -id:A192744 - cp's OEIS frontend

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

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

A192959 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, 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

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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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A192955 Coefficient of x in the reduction by x^2 -> x+1 of the polynomial p(n,x) defined at Comments.

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A192956 Constant term of the reduction by x^2 -> x+1 of the polynomial p(n,x) defined at Comments.

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A192957 Coefficient of x in the reduction by x^2 -> x+1 of the polynomial p(n,x) defined at Comments.

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A192958 Constant term of the reduction by x^2 -> x+1 of the polynomial p(n,x) defined at Comments.

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