A192232 -id:A192232 - cp's OEIS frontend

A192750 Define a pair of sequences c_n, d_n by c_0=0, d_0=1 and thereafter c_n = c_{n-1}+d_{n-1}, d_n = c_{n-1}+4*n+2; sequence here is d_n.

1, 6, 11, 21, 36, 61, 101, 166, 271, 441, 716, 1161, 1881, 3046, 4931, 7981, 12916, 20901, 33821, 54726, 88551, 143281, 231836, 375121, 606961, 982086, 1589051, 2571141, 4160196, 6731341, 10891541, 17622886, 28514431, 46137321, 74651756

Offset: 0

Clark Kimberling, Jul 09 2011

nonn

Old definition was: constant term of the reduction by x^2->x+1 of the polynomial p(n,x) defined recursively by p(n,x) = x*p(n-1,x) + 4n+2 for n>0, with p(0,x)=1.

For discussions of polynomial reduction, see A192232 and A192744.

Harvey P. Dale, Table of n, a(n) for n = 0..1000
Feng-Zhen Zhao, The log-behavior of some sequences related to the generalized Leonardo numbers, Integers (2024) Vol. 24, Art. No. A82.
Index entries for linear recurrences with constant coefficients, signature (2, 0, -1).

See A192751 for c_n.

Cf. A000045, A192744, A192232, A022088, A265750, A265752.

q = x^2; s = x + 1; z = 40;
p[0, n_] := 1; p[n_, x_] := x*p[n - 1, x] + 4 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}]
  (* A192750 *)
u2 = Table[Coefficient[Part[t, n], x, 1], {n, 1, z}]
  (* A192751 *)
LinearRecurrence[{2,0,-1},{1,6,11},40] (* Harvey P. Dale, Dec 03 2023 *)

G.f.: ( 1+4*x-x^2 ) / ( (x-1)*(x^2+x-1) ). The first differences are in A022088. - R. J. Mathar, May 04 2014
a(n) = 5*Fibonacci(n+2)-4. - Gerry Martens, Jul 04 2015
a(n) = A265752(A265750(n)). - Antti Karttunen, Dec 15 2015

Entry revised by N. J. A. Sloane, Dec 15 2015

A192751 Define a pair of sequences c_n, d_n by c_0=0, d_0=1 and thereafter c_n = c_{n-1}+d_{n-1}, d_n = c_{n-1}+4*n+2; sequence here is c_n.

Original entry on oeis.org

0, 1, 7, 18, 39, 75, 136, 237, 403, 674, 1115, 1831, 2992, 4873, 7919, 12850, 20831, 33747, 54648, 88469, 143195, 231746, 375027, 606863, 981984, 1588945, 2571031, 4160082, 6731223, 10891419, 17622760, 28514301, 46137187, 74651618, 120788939

Offset: 0

Clark Kimberling, Jul 09 2011

nonn

Old definition was: coefficient of x in the reduction under x^2->x+1 of the polynomial p(n,x) defined recursively by p(n,x) = x*p(n-1,x) + 4n+2 for n>0, with p(0,x)=1.

For discussions of polynomial reduction, see A192232 and A192744.

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

See A192750 for d_n.

Cf. A000045, A192744, A192232, A022088, A265750, A265752.

(See A192750.)
CoefficientList[Series[x (x^2-4x-1)/((x-1)^2(x^2+x-1)),{x,0,40}],x] (* or *) LinearRecurrence[{3,-2,-1,1},{0,1,7,18},40] (* Harvey P. Dale, Feb 23 2022 *)

G.f.: x*(x^2-4*x-1)/((x-1)^2*(x^2+x-1)). First differences are in A192750. [Colin Barker, Nov 13 2012]
a(n) = 5*Fibonacci(n+3) - (4*n+10). - N. J. A. Sloane, Dec 15 2015
a(n) = A265753(A265750(n)). - Antti Karttunen, Dec 15 2015

Description corrected by Antti Karttunen, Dec 15 2015

Entry revised by N. J. A. Sloane, Dec 15 2015

A192760 Coefficient of x in the reduction by x^2->x+1 of the polynomial p(n,x) defined below in Comments.

Original entry on oeis.org

0, 1, 4, 9, 18, 33, 58, 99, 166, 275, 452, 739, 1204, 1957, 3176, 5149, 8342, 13509, 21870, 35399, 57290, 92711, 150024, 242759, 392808, 635593, 1028428, 1664049, 2692506, 4356585, 7049122, 11405739, 18454894, 29860667, 48315596, 78176299

Offset: 0

Clark Kimberling, Jul 09 2011

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

Form an array with m(1,j) = m(j,1) = j for j >= 1 in the top row and left column, and internal terms m(i,j) = m(i-1,j-1) + m(i-1,j). The sum of the terms in the n-th antidiagonal is a(n). - J. M. Bergot, Nov 07 2012

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

Cf. A192744, A192232.

q = x^2; s = x + 1; z = 40;
p[0, n_] := 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}]  (* A001594 *)
u2 = Table[Coefficient[Part[t, n], x, 1], {n, 1, z}]  (* A192760 *)

a(n) = 2*A000045(n+3)-n-4. G.f. x*(-1-x+x^2) / ( (x^2+x-1)*(x-1)^2 ). - R. J. Mathar, Nov 09 2012

a(n) = Sum_{1..n} C(n-i+2,i+1) + C(n-i,i). - Wesley Ivan Hurt, Sep 13 2017

A192762 Coefficient of x in the reduction by x^2->x+1 of the polynomial p(n,x) defined below in Comments.

Original entry on oeis.org

0, 1, 6, 13, 26, 47, 82, 139, 232, 383, 628, 1025, 1668, 2709, 4394, 7121, 11534, 18675, 30230, 48927, 79180, 128131, 207336, 335493, 542856, 878377, 1421262, 2299669, 3720962, 6020663, 9741658, 15762355, 25504048, 41266439, 66770524

Offset: 0

Clark Kimberling, Jul 09 2011

The titular polynomial is defined recursively by p(n,x)=x*(n-1,x)+n+4 for n>0, where p(0,x)=1. For discussions of polynomial reduction, see A192232 and A192744.

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

Cf. A192744, A192232, A022319 (first differences).

p[0, n_] := 1; p[n_, x_] := x*p[n - 1, x] + n + 4;
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}]
  (* A022319 *)
u2 = Table[Coefficient[Part[t, n], x, 1], {n, 1, z}]
  (* A192762 *)

a(n) = 3*a(n-1)-2*a(n-2)-a(n-3)+a(n-4). G.f.: x*(3*x^2-3*x-1) / ((x-1)^2*(x^2+x-1)). [Colin Barker, Dec 08 2012]

A192883 Constant term in the reduction by (x^2 -> x + 1) of the polynomial F(n+3)*x^n, where F = A000045 (Fibonacci sequence).

Original entry on oeis.org

2, 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, 692290561599

Offset: 0

Clark Kimberling, Jul 12 2011

See A192872.

a(n) is also the area of the triangle with vertices at (F(n),F(n+1)), (F(n+1),F(n)), and (F(n+3),F(n+4)) where F(n) = A000045(n). - J. M. Bergot, May 22 2014

			G.f. = 2 + 5*x^2 + 8*x^3 + 26*x^4 + 63*x^5 + 170*x^6 + ... - _Michael Somos_, Mar 18 2022

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

Cf. A001622, A192232, A192744, A192872.

a:=[2,0,5];; for n in [4..30] do a[n]:=2*a[n-1]+2*a[n-2]-a[n-3]; od; a; # G. C. Greubel, Jan 09 2019

m:=30; R:=PowerSeriesRing(Integers(), m); Coefficients(R!( ( 2-4*x+x^2)/((1+x)*(1-3*x+x^2)) )); // G. C. Greubel, Jan 09 2019

with(combinat):seq(fibonacci(n-1)*fibonacci(n+3), n=0..27): # Gary Detlefs, Oct 19 2011

q = x^2; s = x + 1; z = 28;
p[0, x_] := 2; p[1, x_] := 3 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}] (* A192883 *)
u2 = Table[Coefficient[Part[t, n], x, 1], {n, 1, z}] (* minus A121646 *)
LinearRecurrence[{2,2,-1}, {2,0,5}, 30] (* G. C. Greubel, Jan 09 2019 *)
a[ n_] := Fibonacci[n+1]^2 + (-1)^n; (* Michael Somos, Mar 18 2022 *)

a(n) = round((2^(-1-n)*(7*(-1)^n*2^(1+n)+(3-sqrt(5))^(1+n)+(3+sqrt(5))^(1+n)))/5) \\ Colin Barker, Sep 29 2016

Vec((2+x^2-4*x)/((1+x)*(x^2-3*x+1)) + O(x^40)) \\ Colin Barker, Sep 29 2016

{a(n) = fibonacci(n+1)^2 + (-1)^n}; /* Michael Somos, Mar 18 2022 */

((2-4*x+x^2 )/((1+x)*(1-3*x+x^2))).series(x, 30).coefficients(x, sparse=False) # G. C. Greubel, Jan 09 2019

a(n) = 2*a(n-1) + 2*a(n-2) - a(n-3).
a(n) = Fibonacci(n-1) * Fibonacci(n+3). - Gary Detlefs, Oct 19 2011
a(n) = Fibonacci(n+1)^2 + (-1)^n. - Gary Detlefs, Oct 19 2011
G.f.: ( 2-4*x+x^2 ) / ( (1+x)*(1-3*x+x^2) ). - R. J. Mathar, May 07 2014
a(n) = (2^(-1-n)*(7*(-1)^n*2^(1+n) + (3-sqrt(5))^(1+n) + (3+sqrt(5))^(1+n)))/5. - Colin Barker, Sep 29 2016
From Amiram Eldar, Oct 06 2020: (Start)
Sum_{n>=2} 1/a(n) = 7/18.
Sum_{n>=2} (-1)^n/a(n) = (4/phi - 13/6)/3, where phi is the golden ratio (A001622). (End)
a(n) = a(-2-n) for all n in Z. - Michael Somos, Mar 18 2022

A192914 Constant term in the reduction by (x^2 -> x + 1) of the polynomial C(n)*x^n, where C=A000285.

Original entry on oeis.org

1, 0, 5, 9, 28, 69, 185, 480, 1261, 3297, 8636, 22605, 59185, 154944, 405653, 1062009, 2780380, 7279125, 19057001, 49891872, 130618621, 341963985, 895273340, 2343856029, 6136294753, 16065028224, 42058789925, 110111341545, 288275234716, 754714362597

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. A192232, A192744, A192872.

F:=Fibonacci; List([0..30], n -> F(n+1)^2 +F(n)*F(n-3)); # G. C. Greubel, Jan 12 2019

F:=Fibonacci; [F(n+1)^2+F(n)*F(n-3): n in [0..30]]; // Bruno Berselli, Feb 15 2017

q = x^2; s = x + 1; z = 28;
p[0, x_]:= 1; p[1, x_]:= 4 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 *)
LinearRecurrence[{2,2,-1}, {1,0,5}, 30] (* or *) With[{F:= Fibonacci}, Table[F[n+1]^2 +F[n]*F[n-3], {n, 0, 30}]] (* G. C. Greubel, Jan 12 2019 *)

a(n) = round((2^(-1-n)*(3*(-1)^n*2^(2+n)+(3+sqrt(5))^n*(-1+3*sqrt(5))-(3-sqrt(5))^n*(1+3*sqrt(5))))/5) \\ Colin Barker, Sep 29 2016

Vec((1+3*x^2-2*x)/((1+x)*(x^2-3*x+1)) + O(x^30)) \\ Colin Barker, Sep 29 2016

{f=fibonacci}; vector(30, n, n--; f(n+1)^2 +f(n)*f(n-3)) \\ G. C. Greubel, Jan 12 2019

f=fibonacci; [f(n+1)^2 +f(n)*f(n-3) for n in (0..30)] # G. C. Greubel, Jan 12 2019

a(n) = 2*a(n-1) + 2*a(n-2) - a(n-3).
G.f.: (1 + 3*x^2 - 2*x)/((1 + x)*(x^2 - 3*x + 1)). - R. J. Mathar, May 08 2014
a(n) = (2^(-1-n)*(3*(-1)^n*2^(2+n) + (3 + sqrt(5))^n*(-1 + 3*sqrt(5)) - (3-sqrt(5))^n*(1 + 3*sqrt(5))))/5. - Colin Barker, Sep 29 2016
a(n) = F(n+1)^2 + F(n)*F(n-3). - Bruno Berselli, Feb 15 2017

A192969 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, 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

A049602 a(n) = (Fibonacci(2n)-(-1)^nFibonacci(n))/2.

Original entry on oeis.org

0, 1, 1, 5, 9, 30, 68, 195, 483, 1309, 3355, 8900, 23112, 60813, 158717, 416325, 1088661, 2852242, 7463884, 19546175, 51163695, 133962621, 350695511, 918170280, 2403740304, 6293172025, 16475579353, 43133883845, 112925557953

Offset: 0

Clark Kimberling

A049602 gives the coefficients of x in the reduction of the polynomial p(n,x)=(1/2)((x+1)^n+(x-1)^n) by x^2->x+1. For the constant terms, see A192352. For an introduction to reductions of polynomials by substitutions such as x^2->x+1, see A192232. - Clark Kimberling, Jun 29 2011

Index entries for linear recurrences with constant coefficients, signature (2,3,-4,1).

Cf. A049601.

LinearRecurrence[{2,3,-4,1},{0,1,1,5},30] (* Harvey P. Dale, Jul 07 2017 *)

a(n)=(fibonacci(2*n)-(-1)^n*fibonacci(n))/2 \\ Charles R Greathouse IV, Oct 07 2015

a(n)=Sum{T(2i+1, n-2i-1): i=0, 1, ..., [ (n+1)/2 ]}, array T as in A049600.
Cosh transform of Fibonacci numbers A000045 (or mean of binomial and inverse binomial transforms of A000045). E.g.f.: cosh(x)(2/sqrt(5))*exp(x/2)*sinh(sqrt(5)*x/2). - Paul Barry, May 10 2003
a(n)=sum{k=0..floor(n/2), C(n, 2k)Fib(n-2k)}; - Paul Barry, May 01 2005
a(n)=2a(n-1)+3a(n-2)-4a(n-3)+a(n-4). - Paul Curtz, Jun 16 2008
G.f.: x(1-x)/((1+x-x^2)(1-3x+x^2)); a(n)=sum{k=0..n-1, (-1)^(n-k+1)*F(2k+2)*F(n-k+1)}; - Paul Barry, Jul 11 2008

Simpler description from Vladeta Jovovic and Thomas Baruchel, Aug 24 2004

More terms from Paul Curtz, Jun 16 2008

A086344 a(n) = -2a(n-1) + 4a(n-2), a(0) = 1, a(1) = 0.

Original entry on oeis.org

1, 0, 4, -8, 32, -96, 320, -1024, 3328, -10752, 34816, -112640, 364544, -1179648, 3817472, -12353536, 39976960, -129368064, 418643968, -1354760192, 4384096256, -14187233280, 45910851584, -148570636288, 480784678912, -1555851902976, 5034842521600, -16293092655104

Offset: 0

Paul Barry, Jul 17 2003

Inverse binomial transform of (1,1,5,5,25,25,.....).

The absolute values are the constant terms of the reduction by x^2->x+1 of the polynomial p(n,x) given for d=sqrt(x+1) by p(n,x)=((x+d)^n-(x-d)^n)/(2d), for n>=1. The coefficient of x under this reduction is given by A103435. See A192232 for a discussion of reduction. - Clark Kimberling, Jun 29 2011

Index entries for linear recurrences with constant coefficients, signature (-2,4).

seq((-2)^n * combinat:-fibonacci(n-1), n = 0 .. 100); # Robert Israel, Oct 02 2014

LinearRecurrence[{-2,4},{1,0},40] (* Harvey P. Dale, Oct 10 2018 *)

G.f.: (1+2*x)/((1+(1+sqrt(5))*x)(1+(1-sqrt(5))*x)) = ( -1-2*x ) / ( -1-2*x+4*x^2 ).
E.g.f.: exp(-x)*(cosh(sqrt(5)*x)+sinh(sqrt(5)*x)/sqrt(5)).
a(n)=(sqrt(5)-1)^n*(sqrt(5)/10+1/2)+(-sqrt(5)-1)^n*(1/2-sqrt(5)/10).
(-1)^n*a(n) = A063727(n) - 2*A063727(n-1). - R. J. Mathar, Jul 19 2012
(-1)^n*a(n) = sum(k=0..n, binomial(n,k)*(F(n+1)-F(n))), F(n) Fibonacci number A000045. - Peter Luschny, Oct 01 2014
a(n) = (-2)^n *A000045(n-1). - Robert Israel, Oct 02 2014

A192236 Coefficient of x in the reduction of the n-th 2nd-kind Chebyshev polynomial by x^2 -> x+1.

Original entry on oeis.org

2, 4, 12, 36, 102, 296, 856, 2472, 7146, 20652, 59684, 172492, 498510, 1440720, 4163760, 12033488, 34777426, 100508628, 290475324, 839489268, 2426169014, 7011758584, 20264358408, 58565082744, 169256230458, 489159584636, 1413697437268

Offset: 1

Clark Kimberling, Jun 26 2011

nonn

See A192232.

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

Cf. A192232, A192235, A192237.

a:=[2,4,12,36];; for n in [5..40] do a[n]:=2*a[n-1]+2*a[n-2]+ 2*a[n-3]-a[n-4]; od; a; # G. C. Greubel, Jul 30 2019

I:=[2,4,12,36]; [n le 4 select I[n] else 2*Self(n-1) +2*Self(n-2) +2*Self(n-3) -Self(n-4): n in [1..40]]; // G. C. Greubel, Jul 30 2019

q[x_]:= x + 1; m:=40;
reductionRules = {x^y_?EvenQ -> q[x]^(y/2), x^y_?OddQ -> x q[x]^((y - 1)/2)};
t = Table[Last[Most[FixedPointList[Expand[#1 /. reductionRules] &, ChebyshevU[n, x]]]], {n, m}];
Table[Coefficient[Part[t, n], x, 0], {n, m}] (* A192235 *)
Table[Coefficient[Part[t, n], x, 1], {n, m}] (* A192236 *)
Table[Coefficient[Part[t, n]/2, x, 1], {n, m}] (* A192237 *)
(* Peter J. C. Moses, Jun 25 2011 *)
LinearRecurrence[{2,2,2,-1}, {2,4,12,36}, 40] (* G. C. Greubel, Jul 30 2019 *)

m=40; v=concat([2,4,12,36], vector(m-4)); for(n=5, m, v[n] = 2*v[n-1]+2*v[n-2]+2*v[n-3]-v[n-4]); v \\ G. C. Greubel, Jul 30 2019

def a(n):
    if (n==0): return 2
    elif (1 <= n <= 3): return 4*3^(n-1)
    else: return 2*(a(n-1) + a(n-2) + a(n-3)) - a(n-4)
[a(n) for n in (0..40)] # G. C. Greubel, Jul 30 2019

a(n) = 2*A192237(n+2).

G.f.: 2*x/(1-2*x-2*x^2-2*x^3+x^4). - Colin Barker, Sep 12 2012

cp's OEIS Frontend

A192750 Define a pair of sequences c_n, d_n by c_0=0, d_0=1 and thereafter c_n = c_{n-1}+d_{n-1}, d_n = c_{n-1}+4*n+2; sequence here is d_n.

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A192751 Define a pair of sequences c_n, d_n by c_0=0, d_0=1 and thereafter c_n = c_{n-1}+d_{n-1}, d_n = c_{n-1}+4*n+2; sequence here is c_n.

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A192760 Coefficient of x in the reduction by x^2->x+1 of the polynomial p(n,x) defined below in Comments.

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A192762 Coefficient of x in the reduction by x^2->x+1 of the polynomial p(n,x) defined below in Comments.

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A192883 Constant term in the reduction by (x^2 -> x + 1) of the polynomial F(n+3)*x^n, where F = A000045 (Fibonacci sequence).

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A192914 Constant term in the reduction by (x^2 -> x + 1) of the polynomial C(n)*x^n, where C=A000285.

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

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A049602 a(n) = (Fibonacci(2n)-(-1)^nFibonacci(n))/2.

A086344 a(n) = -2a(n-1) + 4a(n-2), a(0) = 1, a(1) = 0.