A192872 -id:A192872 - cp's OEIS frontend

A194038 Natural interspersion of A034856, a rectangular array, by antidiagonals.

1, 4, 2, 8, 5, 3, 13, 9, 6, 7, 19, 14, 10, 11, 12, 26, 20, 15, 16, 17, 18, 34, 27, 21, 22, 23, 24, 25, 43, 35, 28, 29, 30, 31, 32, 33, 53, 44, 36, 37, 38, 39, 40, 41, 42, 64, 54, 45, 46, 47, 48, 49, 50, 51, 52, 76, 65, 55, 56, 57, 58, 59, 60, 61, 62, 63, 89, 77, 66

Offset: 0

Clark Kimberling, Aug 12 2011

See A194029 for definitions of natural fractal sequence and natural interspersion. Every positive integer occurs exactly once (and every pair of rows intersperse), so that as a sequence, A194038 is a permutation of the positive integers; its inverse is A194040.

			Northwest corner:
1...4...8...13...19
2...5...9...14...20
3...6...10..15...21
7...11..16..22...29
12..17..23..30...38

Cf. A192872, A194040.

z = 30;
c[k_] := (k^2 + 3 k - 2)/2;
c = Table[c[k], {k, 1, z}]  (* A034856 *)
f[n_] := If[MemberQ[c, n], 1, 1 + f[n - 1]]
f = Table[f[n], {n, 1, 255}]  (* essentially A002260 *)
r[n_] := Flatten[Position[f, n]]
t[n_, k_] := r[n][[k]]
TableForm[Table[t[n, k], {n, 1, 7}, {k, 1, 7}]]
p = Flatten[Table[t[k, n - k + 1], {n, 1, 13}, {k, 1, n}]]   (* A194038 *)
q[n_] := Position[p, n]; Flatten[Table[q[n], {n, 1, 70}]]  (* A194040 *)

A192906 Constant term in the reduction by (x^2 -> x + 1) of the polynomial p(n,x) defined below at Comments.

Original entry on oeis.org

1, 1, 2, 7, 23, 72, 225, 705, 2210, 6927, 21711, 68048, 213281, 668481, 2095202, 6566935, 20582567, 64511384, 202196289, 633738369, 1986309058, 6225634847, 19512839199, 61158565024

Offset: 0

Clark Kimberling, Jul 12 2011

The titular polynomial is defined by p(n,x) = (x^2)*p(n-1,x) + x*p(n-2,x), with p(0,x) = 1, p(1,x) = x + 1.

G. C. Greubel, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (3,0,1,1).

Cf. A192232, A192744, A192872, A192904, A192907.

a:=[1,1,2,7];; for n in [5..30] do a[n]:=3*a[n-1]+a[n-3]+a[n-4]; od; a; # G. C. Greubel, Jan 11 2019

m:=30; R:=PowerSeriesRing(Integers(), m); Coefficients(R!( (1-2*x-x^2)/(1-3*x-x^3-x^4) )); // G. C. Greubel, Jan 11 2019

(* To obtain very general results, delete the next line. *)
u = 1; v = 1; a = 1; b = 1; c = 1; d = 1; e = 1; f = 0;
q = x^2; s = u*x + v; z = 24;
p[0, x_] := a;
p[1, x_] := b*x + c; p[n_, x_] :=  d*(x^2)*p[n - 1, x] + e*x*p[n - 2, x] + f; Table[Expand[p[n, x]], {n, 0, 8}] (* p(0,x), p(1,x), ... p(5,x) *)
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}];
u0 = Table[Coefficient[Part[t, n], x, 0], {n, 1, z}]    (* A192904 *)
u1 = Table[Coefficient[Part[t, n], x, 1], {n, 1, z}]    (* A192905 *)
Simplify[FindLinearRecurrence[u0]] (* recurrence for 0-sequence *)
Simplify[FindLinearRecurrence[u1]] (* recurrence for 1-sequence *)
LinearRecurrence[{3,0,1,1}, {1,1,2,7}, 30] (* G. C. Greubel, Jan 11 2019 *)

my(x='x+O('x^30)); Vec((1-2*x-x^2)/(1-3*x-x^3-x^4)) \\ G. C. Greubel, Jan 11 2019

((1-2*x-x^2)/(1-3*x-x^3-x^4)).series(x, 30).coefficients(x, sparse=False) # G. C. Greubel, Jan 11 2019

a(n) = 3*a(n-1) + a(n-3) + a(n-4).

G.f.: (1-2*x-x^2)/(1-3*x-x^3-x^4). - Colin Barker, Aug 31 2012

A192916 Constant term in the reduction by (x^2 -> x+1) of the polynomial C(n)*x^n, where C=A022095.

Original entry on oeis.org

1, 0, 6, 11, 34, 84, 225, 584, 1534, 4011, 10506, 27500, 72001, 188496, 493494, 1291979, 3382450, 8855364, 23183649, 60695576, 158903086, 416013675, 1089137946, 2851400156, 7465062529, 19543787424, 51166299750, 133955111819, 350699035714, 918141995316

Offset: 0

Clark Kimberling, Jul 12 2011

See A192872.

Colin Barker, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (2,2,-1).

Cf. A000045, A192232, A192744, A192872, A192917.

F:=Fibonacci;; List([0..30], n-> F(2*n) +2*F(n)*F(n-1) +(-1)^n); # G. C. Greubel, Jul 28 2019

F:=Fibonacci; [F(2*n) +2*F(n)*F(n-1) +(-1)^n: n in [0..30]]; // G. C. Greubel, Jul 28 2019

(* First program *)
q = x^2; s = x + 1; z = 28;
p[0, x_]:= 1; p[1, x_]:= 5 x;
p[n_, x_]:= p[n-1, x]*x + p[n-2, x]*x^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}] (* A192914 *)
u2 = Table[Coefficient[Part[t, n], x, 1], {n, 1, z}] (* see A192878 *)
(* Second program *)
With[{F=Fibonacci}, Table[F[2*n] +2*F[n]*F[n-1] +(-1)^n, {n,0,30}]] (* G. C. Greubel, Jul 28 2019 *)

a(n) = round((2^(-n)*(7*(-2)^n+(3+sqrt(5))^n*(-1+2*sqrt(5))-(3-sqrt(5))^n*(1+2*sqrt(5))))/5) \\ Colin Barker, Oct 01 2016

Vec((1+4*x^2-2*x)/((1+x)*(1-3*x+x^2)) + O(x^30)) \\ Colin Barker, Oct 01 2016

vector(30, n, n--; f=fibonacci; f(2*n) +2*f(n)*f(n-1) +(-1)^n) \\ G. C. Greubel, Jul 28 2019

f=fibonacci; [f(2*n) +2*f(n)*f(n-1) +(-1)^n for n in (0..30)] # G. C. Greubel, Jul 28 2019

a(n) = 2*a(n-1) + 2*a(n-2) - a(n-3).
G.f.: (1 -2*x +4*x^2)/((1+x)*(1-3*x+x^2)). - R. J. Mathar, May 08 2014
a(n) + a(n+1) = A054492(n). - R. J. Mathar, May 07 2014
a(n) = (2^(-n)*(7*(-2)^n+(3+sqrt(5))^n*(-1+2*sqrt(5))-(3-sqrt(5))^n*(1+2*sqrt(5))))/5. - Colin Barker, Oct 01 2016
a(n) = Fibonacci(2*n) + 2*Fibonacci(n)*Fibonacci(n-1) + (-1)^n. - G. C. Greubel, Jul 28 2019

A192917 Coefficient of x in the reduction by (x^2 -> x+1) of the polynomial C(n)*x^n, where C=A022095.

Original entry on oeis.org

0, 5, 6, 22, 51, 140, 360, 949, 2478, 6494, 16995, 44500, 116496, 304997, 798486, 2090470, 5472915, 14328284, 37511928, 98207509, 257110590, 673124270, 1762262211, 4613662372, 12078724896, 31622512325, 82788812070, 216743923894, 567442959603, 1485584954924

Offset: 0

Clark Kimberling, Jul 12 2011

See A192872.

Colin Barker, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (2,2,-1).

Cf. A000045, A192232, A192744, A192872, A192916.

F:=Fibonacci;; List([0..30], n-> F(2*n+1) +2*F(n)^2 -(-1)^n); # G. C. Greubel, Jul 29 2019

F:=Fibonacci; [F(2*n+1) +2*F(n)^2 -(-1)^n: n in [0..30]]; // G. C. Greubel, Jul 29 2019

(* First program *)
q = x^2; s = x + 1; z = 28;
p[0, x_]:= 1; p[1, x_]:= 5 x;
p[n_, x_]:= p[n-1, x]*x + p[n-2, x]*x^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}] (* A192914 *)
u2 = Table[Coefficient[Part[t, n], x, 1], {n, 1, z}] (* see A192878 *)
(* Second program *)
With[{F=Fibonacci}, Table[F[2*n+1] +2*F[n]^2 -(-1)^n, {n,0,30}]] (* G. C. Greubel, Jul 28 2019 *)

a(n) = round((2^(-1-n)*(-9*(-1)^n*2^(1+n)-(3-sqrt(5))^n*(-9+sqrt(5))+(3+sqrt(5))^n*(9+sqrt(5))))/5) \\ Colin Barker, Oct 01 2016

concat(0, Vec((-x*(-5+4*x))/((1+x)*(x^2-3*x+1)) + O(x^40))) \\ Colin Barker, Oct 01 2016

vector(30, n, n--; f=fibonacci; f(2*n+1) +2*f(n)^2 -(-1)^n) \\ G. C. Greubel, Jul 29 2019

f=fibonacci; [f(2*n+1) +2*f(n)^2 -(-1)^n for n in (0..30)] # G. C. Greubel, Jul 29 2019

a(n) = 2*a(n-1) + 2*a(n-2) - a(n-3).
G.f.: x*(5 -4*x)/((1+x)*(1-3*x+x^2)). - R. J. Mathar, May 08 2014
a(n) = -3*a(n-1) +a(n-2) = 9*(-1)^(n+1). - R. J. Mathar, May 08 2014
a(n) = (2^(-1-n)*(-9*(-1)^n*2^(1+n)-(3-sqrt(5))^n*(-9+sqrt(5))+(3+sqrt(5))^n*(9+sqrt(5))))/5. - Colin Barker, Oct 01 2016
a(n) = Fibonacci(2*n+1) + 2*Fibonacci(n)^2 - (-1)^n. - G. C. Greubel, Jul 29 2019

A192919 Constant term in the reduction by (x^2 -> x+1) of the polynomial F(n+4)*x^n, where F=A000045 (Fibonacci sequence).

Original entry on oeis.org

3, 0, 8, 13, 42, 102, 275, 712, 1872, 4893, 12818, 33550, 87843, 229968, 602072, 1576237, 4126650, 10803702, 28284467, 74049688, 193864608, 507544125, 1328767778, 3478759198, 9107509827, 23843770272, 62423801000, 163427632717, 427859097162, 1120149658758

Offset: 0

Clark Kimberling, Jul 12 2011

See A192872.

Colin Barker, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (2,2,-1).

Cf. A000045, A001622, A192232, A192744, A192872, A192883, A192920, A290565.

F:=Fibonacci;; List([0..30], n-> F(n-1)*F(n+4)); # G. C. Greubel, Jul 28 2019

F:=Fibonacci; [F(n-1)*F(n+4): n in [0..30]]; // G. C. Greubel, Jul 28 2019

with(combinat):seq(fibonacci(n-1)*fibonacci(n+4), n=0..27);

(* First program *)
q = x^2; s = x + 1; z = 28;
p[0, x_]:= 3; p[1, x_]:= 5 x;
p[n_, x_]:= p[n-1, x]*x + p[n-2, x]*x^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}] (* A192919 *)
u2 = Table[Coefficient[Part[t, n], x, 1], {n, 1, z}] (* A192920 *)
(* Second program *)
With[{F=Fibonacci}, Table[F[n-1]*F[n+4], {n,0,30}]] (* G. C. Greubel, Jul 28 2019 *)

a(n) = round((2^(-n)*(11*(-2)^n-(3-sqrt(5))^n*(-2+sqrt(5))+(2+sqrt(5))*(3+sqrt(5))^n))/5) \\ Colin Barker, Oct 01 2016

Vec((3+2*x^2-6*x)/((1+x)*(x^2-3*x+1)) + O(x^30)) \\ Colin Barker, Oct 01 2016

vector(30, n, n--; f=fibonacci; f(n-1)*f(n+4)) \\ G. C. Greubel, Jul 28 2019

f=fibonacci; [f(n-1)*f(n+4) for n in (0..30)] # G. C. Greubel, Jul 28 2019

a(n) = 2*a(n-1) + 2*a(n-2) - a(n-3).
a(n) = Fibonacci(n-1)*Fibonacci(n+4). - Gary Detlefs, Oct 19 2011
G.f.: (3 -6*x +2*x^2)/((1+x)*(1-3*x+x^2)). - R. J. Mathar, May 08 2014
a(n) + a(n+1) = A001906(n+1). - R. J. Mathar, May 08 2014
a(n) = (2^(-n)*(11*(-2)^n-(3-sqrt(5))^n*(-2+sqrt(5))+(2+sqrt(5))*(3+sqrt(5))^n))/5. - Colin Barker, Oct 01 2016
From Amiram Eldar, Oct 06 2020: (Start)
Sum_{n>=2} 1/a(n) = (1/5) * A290565 - 17/150.
Sum_{n>=2} (-1)^n/a(n) = 1/phi - 83/150, where phi is the golden ratio (A001622). (End)

A192920 Coefficient of x in the reduction by (x^2 -> x+1) of the polynomial F(n+4)*x^n, where F=A000045 (Fibonacci sequence).

Original entry on oeis.org

0, 5, 8, 26, 63, 170, 440, 1157, 3024, 7922, 20735, 54290, 142128, 372101, 974168, 2550410, 6677055, 17480762, 45765224, 119814917, 313679520, 821223650, 2149991423, 5628750626, 14736260448, 38580030725, 101003831720, 264431464442

Offset: 0

Clark Kimberling, Jul 12 2011

nonn

See A192872.

G. C. Greubel, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (2,2,-1).

Cf. A000045, A192232, A192744, A192872, A192883, A192919.

a:=[0,5,8];; for n in [4..30] do a[n]:=2*a[n-1]+2*a[n-2]-a[n-3]; od; a; # G. C. Greubel, Feb 06 2019

[Fibonacci(n+2)^2 -(-1)^n: n in [0..30]]; // G. C. Greubel, Feb 06 2019, modified Jul 28 2019

(* First program *)
q = x^2; s = x + 1; z = 28;
p[0, x_]:= 3; p[1, x_]:= 5 x;
p[n_, x_]:= p[n-1, x]*x + p[n-2, x]*x^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}] (* A192919 *)
u2 = Table[Coefficient[Part[t, n], x, 1], {n, 1, z}] (* A192920 *)
(* Second program *)
LinearRecurrence[{2,2,-1}, {0,5,8}, 30] (* G. C. Greubel, Feb 06 2019 *)

vector(30, n, n--; fibonacci(n+2)^2 -(-1)^n) \\ G. C. Greubel, Feb 06 2019, modified Jul 28 2019

[fibonacci(n+2)^2 -(-1)^n for n in (0..30)] # G. C. Greubel, Feb 06 2019, modified Jul 28 2019

a(n) = 2*a(n-1) + 2*a(n-2) - a(n-3).
a(n) = A192883(n+1).
G.f.: x*(5-2*x)/((1+x)*(1-3*x+x^2)). - R. J. Mathar, Aug 01 2011
a(n) = (A005248(n+2) - 7*(-1)^n)/5. - R. J. Mathar, Aug 01 2011
a(n) = Fibonacci(n+2)^2 - (-1)^n. - G. C. Greubel, Feb 06 2019
Sum_{n>=1} 1/a(n) = 7/18. - Amiram Eldar, Oct 05 2020

A192922 Constant term in the reduction by (x^2->x+1) of the polynomial p(n,x) defined below at Comments.

Original entry on oeis.org

1, 0, 1, 2, 5, 11, 25, 55, 122, 268, 590, 1295, 2844, 6240, 13693, 30039, 65900, 144559, 317108, 695595, 1525829, 3346965, 7341695, 16104238, 35325142, 77486710, 169969295, 372832346

Offset: 0

Clark Kimberling, Jul 12 2011

nonn

The titular polynomial is defined by p(n,x) = p(n-1,x) +(x^2)*p(n-2,x), with p(0,x)=1, p(1,x)=x. For discussions of polynomial reduction, see A192232, A192744, and A192872.

G. C. Greubel, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (2,2,-3,-1).

Cf. A192232, A192744, A192923.

a:=[1,0,1,2];; for n in [5..30] do a[n]:=2*a[n-1]+2*a[n-2] -3*a[n-3]-a[n-4]; od; a; # G. C. Greubel, Feb 06 2019

m:=30; R:=PowerSeriesRing(Integers(), m); Coefficients(R!( (1-x^2-2*x+3*x^3)/(1-2*x-2*x^2+3*x^3+x^4) )); // G. C. Greubel, Feb 06 2019

q = x^2; s = x + 1; z = 28;
p[0, x_] := 1; p[1, x_] := x;
p[n_, x_] := p[n - 1, x] + p[n - 2, x]*x^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}]
  (* A192922 *)
u2 = Table[Coefficient[Part[t, n], x, 1], {n, 1, z}]  (* A192923 *)
LinearRecurrence[{2,2,-3,-1}, {1,0,1,2}, 30] (* G. C. Greubel, Feb 06 2019 *)

my(x='x+O(x^30)); Vec((1-x^2-2*x+3*x^3)/(1-2*x-2*x^2+3*x^3+x^4)) \\ G. C. Greubel, Feb 06 2019

((1-x^2-2*x+3*x^3)/(1-2*x-2*x^2+3*x^3+x^4)).series(x, 30).coefficients(x, sparse=False)

a(n) = 2*a(n-1) + 2*a(n-2) - 3*a(n-3) - a(n-4).

G.f.: (1-x^2-2*x+3*x^3)/(1-2*x-2*x^2+3*x^3+x^4). - R. J. Mathar, May 08 2014

A192923 Coefficient of x in the reduction by (x^2->x+1) of the polynomial p(n,x) defined below at Comments.

Original entry on oeis.org

0, 1, 2, 4, 9, 19, 42, 91, 200, 437, 959, 2101, 4609, 10106, 22168, 48620, 106649, 233928, 513126, 1125541, 2468901, 5415578, 11879209, 26057330, 57157443, 125376341, 275016369, 603255761

Offset: 0

Clark Kimberling, Jul 12 2011

nonn

The titular polynomial is defined by p(n,x) = p(n-1,x) +(x^2)*p(n-2,x), with p(0,x)=1, p(1,x)=x. For discussions of polynomial reduction, see A192232, A192744, and A192872.

G. C. Greubel, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (2,2,-3,-1).

Cf. A192232, A192744, A192922.

a:=[0,1,2,4];; for n in [5..30] do a[n]:=2*a[n-1]+2*a[n-2]-3*a[n-3] -a[n-4]; od; a; # G. C. Greubel, Feb 06 2019

m:=30; R:=PowerSeriesRing(Integers(), m); [0] cat Coefficients(R!( x*(1-2*x^2)/(1-2*x-2*x^2+3*x^3+x^4) )); // G. C. Greubel, Feb 06 2019

(See A192922.)
CoefficientList[Series[x*(1-2*x^2)/(1-2*x-2*x^2+3*x^3+x^4), {x, 0, 30}], x] (* G. C. Greubel, Jun 26 2017 *)

x='x+O('x^30); concat([0], Vec(x*(1-2*x^2)/(1-2*x-2*x^2+3*x^3+x^4) )) \\ G. C. Greubel, Jun 26 2017

(x*(1-2*x^2)/(1-2*x-2*x^2+3*x^3+x^4)).series(x, 30).coefficients(x, sparse=False) # G. C. Greubel, Feb 06 2019

a(n) = 2*a(n-1) + 2*a(n-2) - 3*a(n-3) - a(n-4).

G.f.: x*(1-2*x^2) / ( 1-2*x-2*x^2+3*x^3+x^4 ). - R. J. Mathar, May 08 2014

A192924 Constant term in the reduction by (x^2->x+1) of the polynomial p(n,x) defined below at Comments.

Original entry on oeis.org

1, 0, 1, 4, 20, 115, 761, 5723, 48353, 454233, 4701724, 53204955, 653749199, 8670930456, 123500484305, 1880367585200, 30481594476514, 524197712831867, 9532792177527307, 182792169717039937, 3686148742978363201, 77989408383978583425

Offset: 0

Clark Kimberling, Jul 12 2011

nonn

The titular polynomial is defined by p(n,x)=n*p(n-1,x)+(x^2)*p(n-2,x), with p(0,x)=1, p(1,x)=x. For discussions of polynomial reduction, see A192232, A192744, and A192872.

Cf. A192232, A192744, A192925.

q = x^2; s = x + 1; z = 22;
p[0, x_] := 1; p[1, x_] := x;
p[n_, x_] := n*p[n - 1, x] + p[n - 2, x]*x^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}]
  (* A192924 *)
u2 = Table[Coefficient[Part[t, n], x, 1], {n, 1, z}]
  (* A192925 *)

Conjecture: a(n) +2*(-n+1)*a(n-1) +(n^2-3*n-1)*a(n-2) +3*(n-2)*a(n-3) +a(n-4)=0. - R. J. Mathar, May 08 2014

A192921 Constant term in the reduction by (x^2->x+1) of the polynomial p(n,x) defined below at Comments.

Original entry on oeis.org

1, 2, 2, 7, 16, 44, 113, 298, 778, 2039, 5336, 13972, 36577, 95762, 250706, 656359, 1718368, 4498748, 11777873, 30834874, 80726746, 211345367, 553309352, 1448582692, 3792438721, 9928733474, 25993761698, 68052551623, 178163893168, 466439127884, 1221153490481

Offset: 0

Clark Kimberling, Jul 12 2011

The titular polynomial is defined by p(n,x) = x*p(n-1,x) +(x^2)*p(n-2,x), with p(0,x)=1, p(1,x)=1+x^2. For discussions of polynomial reduction, see A192232, A192744, and A192872.

			The coefficients in the polynomials p(n,x) are Fibonacci numbers.  The first seven and their reductions:
...
1 -> 1
1 + x^2 -> 2 + x
x + x^2 + x^3 -> 2 + 4*x
2*x^2 + x^3 + 2*x^4 -> 7 + 10*x
3*x^3 + 2*x^4 + 3*x^5 -> 16 + 27*x
5*x^4 + 3*x^5 + 5*x^6 -> 44 + 70*x
8*x^5 + 5*x^6 + 8*x^7 -> 113 + 184*x,
so that A192921=(1,2,2,7,16,44,113,...).

Alois P. Heinz, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (2,2,-1).

Cf. A192232, A192744, A192872.

List([0..30], n -> Fibonacci(n-2)^2 +Fibonacci(n)*Fibonacci(n+1)); # G. C. Greubel, Feb 06 2019

[Fibonacci(n-2)^2 + Fibonacci(n)*Fibonacci(n+1): n in [0..30]]; // G. C. Greubel, Feb 06 2019

a:= n-> (<<0|1|0>, <0|0|1>, <-1|2|2>>^n. <<1,2,2>>)[1,1]:
seq(a(n), n=0..30);  # Alois P. Heinz, Sep 28 2016

q = x^2; s = x + 1; z = 28;
p[0, x_] := 1; p[1, x_] := x^2 + 1;
p[n_, x_] := p[n - 1, x]*x + p[n - 2, x]*x^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}]
  (* A192921 *)
u2 = Table[Coefficient[Part[t, n], x, 1], {n, 1, z}]
  (* A192879 *)
LinearRecurrence[{2,2,-1}, {1,2,2}, 30] (* G. C. Greubel, Feb 06 2019 *)

a(n) = round((2^(-n)*(-3*(-2)^n-(-4+sqrt(5))*(3+sqrt(5))^n+(3-sqrt(5))^n*(4+sqrt(5))))/5) \\ Colin Barker, Oct 01 2016

Vec(-(2*x-1)*(1+2*x)/((1+x)*(x^2-3*x+1)) + O(x^40)) \\ Colin Barker, Oct 01 2016

{a(n) = fibonacci(n-2)^2 +fibonacci(n)*fibonacci(n+1)}; \\ G. C. Greubel, Feb 06 2019

[fibonacci(n-2)^2 +fibonacci(n)*fibonacci(n+1) for n in range(30)] # G. C. Greubel, Feb 06 2019

a(n) = 2*a(n-1) + 2*a(n-2) - a(n-3).
G.f.: (1-2*x)*(1+2*x) / ( (1+x)*(1-3*x+x^2) ). - R. J. Mathar, May 08 2014
a(n) = A059929(n-1) + 2*A059929(n-2). - R. J. Mathar, May 08 2014
a(n) = F(n-4)*F(n) + F(n-1)*F(n+2), where F(-4)=-3, F(-3)=2, F(-2)=-1, F(-1)=1. - Bruno Berselli, Nov 03 2015
a(n) = (2^(-n)*(-3*(-2)^n-(-4+sqrt(5))*(3+sqrt(5))^n+(3-sqrt(5))^n*(4+sqrt(5))))/5. - Colin Barker, Oct 01 2016

cp's OEIS Frontend

A194038 Natural interspersion of A034856, a rectangular array, by antidiagonals.

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A192906 Constant term in the reduction by (x^2 -> x + 1) of the polynomial p(n,x) defined below at Comments.

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A192916 Constant term in the reduction by (x^2 -> x+1) of the polynomial C(n)*x^n, where C=A022095.

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A192917 Coefficient of x in the reduction by (x^2 -> x+1) of the polynomial C(n)*x^n, where C=A022095.

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A192919 Constant term in the reduction by (x^2 -> x+1) of the polynomial F(n+4)*x^n, where F=A000045 (Fibonacci sequence).

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A192920 Coefficient of x in the reduction by (x^2 -> x+1) of the polynomial F(n+4)*x^n, where F=A000045 (Fibonacci sequence).

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