A192232 -id:A192232 - cp's OEIS frontend

A265398 Perform one x^2 -> x+1 reduction for the polynomial with nonnegative integer coefficients that is encoded in the prime factorization of n.

1, 2, 3, 4, 6, 6, 15, 8, 9, 12, 35, 12, 77, 30, 18, 16, 143, 18, 221, 24, 45, 70, 323, 24, 36, 154, 27, 60, 437, 36, 667, 32, 105, 286, 90, 36, 899, 442, 231, 48, 1147, 90, 1517, 140, 54, 646, 1763, 48, 225, 72, 429, 308, 2021, 54, 210, 120, 663, 874, 2491, 72, 3127, 1334, 135, 64, 462, 210, 3599, 572, 969, 180, 4087, 72

Offset: 1

Antti Karttunen, Dec 15 2015

Completely multiplicative with a(2) = 2, a(3) = 3, a(prime(k)) = prime(k-1) * prime(k-2) for k > 2. - Andrew Howroyd & Antti Karttunen, Aug 04 2018

Antti Karttunen, Table of n, a(n) for n = 1..10080

Cf. A064989, A065330, A065331.

Cf. also A192232, A206296, A265399.

a[n_] := a[n] = Module[{k, p, e}, Which[n<4, n, PrimeQ[n], k = PrimePi[n]; Prime[k-1] Prime[k-2], True, Product[{p, e} = pe; a[p]^e, {pe, FactorInteger[n]}]]];
a /@ Range[1, 72] (* Jean-François Alcover, Sep 20 2019 *)
f[p_, e_] := If[p < 5, p, NextPrime[p,-1]*NextPrime[p,-2]]^e; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Dec 01 2022 *)

A065330(n) = { while(0 == (n%2), n = n/2); while(0 == (n%3), n = n/3); n; }
A065331 = n -> n/A065330(n);
A064989(n) = {my(f); f = factor(n); if((n>1 && f[1,1]==2), f[1,2] = 0); for (i=1, #f~, f[i,1] = precprime(f[i,1]-1)); factorback(f)};
A265398(n) = { my(a); if(1 == n, n, a = A064989(A065330(n)); A064989(a)*a*A065331(n)); };

r(p) = {my(q = precprime(p-1)); q*precprime(q-1)};
a(n) = {my(f=factor(n)); prod(i=1, #f~, if(f[i,1]<5, f[i,1], r(f[i,1]))^f[i,2])}; \\ Amiram Eldar, Dec 01 2022

(definec (A265398 n) (if (= 1 n) n (* (A065331 n) (A064989 (A065330 n)) (A064989 (A064989 (A065330 n))))))

a(1) = 1; for n > 1, a(n) = A064989(A064989(A065330(n))) * A064989(A065330(n)) * A065331(n).

Sum_{k=1..n} a(k) = c * n^3, where c = (1/3) * Product_{p prime} (p^3-p^2)/(p^3-a(p)) = 0.093529982... . - Amiram Eldar, Dec 01 2022

Keyword mult added by Antti Karttunen, Aug 04 2018

A265399 Repeatedly perform x^2 -> x+1 reduction for polynomial (with nonnegative integer coefficients) encoded in prime factorization of n, until the polynomial is at most degree 1.

Original entry on oeis.org

1, 2, 3, 4, 6, 6, 18, 8, 9, 12, 108, 12, 1944, 36, 18, 16, 209952, 18, 408146688, 24, 54, 216, 85691213438976, 24, 36, 3888, 27, 72, 34974584955819144511488, 36, 2997014624388697307377363936018956288, 32, 324, 419904, 108, 36, 104819342594514896999066634490728502944926883876041385836544, 816293376, 5832, 48

Offset: 1

Antti Karttunen, Dec 15 2015

In terms of integers: apply A265398 as many times as necessary to n, until it gets 3-smooth, one of the terms of A003586.

Completely multiplicative with a(2) = 2, a(3) = 3, a(p) = a(A265398(p)) for p > 3. - Andrew Howroyd & Antti Karttunen, Aug 04 2018

Antti Karttunen, Table of n, a(n) for n = 1..60

Cf. A003586 (fixed points), A065331.

Cf. also A192232, A206296, A265398, A265752, A265753.

f[p_, e_] := If[p < 5, p, a[NextPrime[p, -1] * NextPrime[p, -2]]]^e; a[1] = 1; a[n_] := a[n] = Times @@ f @@@ FactorInteger[n]; Array[a, 40] (* Amiram Eldar, Sep 07 2023 *)

\\ Needs also code from A265398.
A265399(n) = if(A065331(n) == n, n, A265399(A265398(n)));
for(n=1, 60, write("b265399.txt", n, " ", A265399(n)));

(definec (A265399 n) (if (= (A065331 n) n) n (A265399 (A265398 n))))

If A065331(n) = n [that is, when n is one of 3-smooth numbers, A003586] then a(n) = n, otherwise a(n) = a(A265398(n)).
Other identities. For all n >= 1:
a(n) = 2^A265752(n) * 3^A265753(n).

Keyword mult added by Antti Karttunen, Aug 04 2018

A192234 a(n) = 2*(a(n-1) + a(n-2) + a(n-3)) - a(n-4) for n >= 4, with initial terms 0,1,0,1.

Original entry on oeis.org

0, 1, 0, 1, 4, 9, 28, 81, 232, 673, 1944, 5617, 16236, 46921, 135604, 391905, 1132624, 3273345, 9460144, 27340321, 79014996, 228357577, 659965644, 1907336113, 5512303672, 15930853281, 46041020488, 133061018769, 384553481404, 1111380188041

Offset: 0

Clark Kimberling, Jun 26 2011

With a different offset, constant term of the reduction of the n-th 1st-kind Chebyshev polynomial by x^2->x+1. See A192232.

Colin Barker, Table of n, a(n) for n = 0..1000
H. S. M. Coxeter, Loxodromic sequences of tangent spheres, Aequationes Mathematicae, 1.1-2 (1968): 104-121. See p. 112.
Index entries for linear recurrences with constant coefficients, signature (2,2,2,-1).

Cf. A192232.

a:=[0,1,0,1];; 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 29 2019

R:=PowerSeriesRing(Integers(), 40); [0] cat Coefficients(R!( x*(1-2*x-x^2)/(1-2*x-2*x^2-2*x^3+x^4) )); // G. C. Greubel, Jul 29 2019

q[x_]:= x + 1;
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] &, ChebyshevT[n, x]]]], {n, 1, 40}];
Table[Coefficient[Part[t, n], x, 0], {n, 1, 40}]  (* A192234 *)
Table[Coefficient[Part[t, n], x, 1], {n, 1, 40}]  (* A071101 *)
(* Peter J. C. Moses, Jun 25 2011 *)

a(n)=my(t=polchebyshev(n));while(poldegree(t)>1, t=substpol(t, x^2,x+1));subst(t,x,0) \\ Charles R Greathouse IV, Feb 09 2012

concat(0, Vec(x*(1-2*x-x^2)/(1-2*x-2*x^2-2*x^3+x^4) + O(x^40))) \\ Colin Barker, Sep 09 2018

(x*(1-2*x-x^2)/(1-2*x-2*x^2-2*x^3+x^4)).series(x, 40).coefficients(x, sparse=False) # G. C. Greubel, Jul 29 2019

G.f.: x*(1 - 2*x - x^2) / (1 - 2*x - 2*x^2 - 2*x^3 + x^4). - Colin Barker, Feb 09 2012 and Sep 09 2018

Entry revised (with new offset and initial terms) by N. J. A. Sloane, Sep 03 2018

A192235 Constant term of the reduction of the n-th 2nd-kind Chebyshev polynomial by x^2 -> x+1.

Original entry on oeis.org

0, 3, 8, 21, 64, 183, 528, 1529, 4416, 12763, 36888, 106605, 308096, 890415, 2573344, 7437105, 21493632, 62117747, 179523624, 518832901, 1499454912, 4333505127, 12524062256, 36195211689, 104606103232, 302317249227, 873713066040

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, A192236, A192237.

a:=[0,3,8,21];; 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:=[0, 3, 8, 21]; [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;
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, 1, 40}];
Table[Coefficient[Part[t, n], x, 0], {n, 1, 40}] (* A192235 *)
Table[Coefficient[Part[t, n], x, 1], {n, 1, 40}] (* A192236 *)
Table[Coefficient[Part[t, n]/2, x, 1], {n, 1, 40}] (* A192237 *)
(* by Peter J. C. Moses, Jun 25 2011 *)
LinearRecurrence[{2,2,2,-1}, {0,3,8,21}, 40] (* G. C. Greubel, Jul 30 2019 *)

a(n)=my(t=polchebyshev(n,2));while(poldegree(t)>1, t=substpol(t, x^2,x+1));subst(t,x,0) \\ Charles R Greathouse IV, Feb 09 2012

m=40; v=concat([0, 3, 8, 21], 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

@cached_function
def a(n):
    if (n==0): return 0
    elif (1 <= n <= 3): return fibonacci(2*n+2)
    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

Empirical G.f.: x^2*(3-x)*(1+x)/(1-2*x-2*x^2-2*x^3+x^4). - Colin Barker, Sep 11 2012

A192237 a(n) = 2*(a(n-1) + a(n-2) + a(n-3)) - a(n-4) for n >= 4, with initial terms 0,0,0,1.

Original entry on oeis.org

0, 0, 0, 1, 2, 6, 18, 51, 148, 428, 1236, 3573, 10326, 29842, 86246, 249255, 720360, 2081880, 6016744, 17388713, 50254314, 145237662, 419744634, 1213084507, 3505879292, 10132179204, 29282541372, 84628115229, 244579792318, 706848718634, 2042830710990, 5903890328655, 17062559724240, 49311712809136, 142513495013072

Offset: 0

Clark Kimberling, Jun 26 2011

nonn

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

Cf. A192232, A192235.
With a different offset, equals (A192236)/2.
Other sequences with this recurrence but different initial conditions: A192234, A317973, A317974, A317975, A317976.

a:=[0,0,0,1];; 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:=[0,0,0,1]; [n le 4 select I[n] else 2*(Self(n-1)+Self(n-2) +Self(n-3))-Self(n-4): n in [1..40]]; // Vincenzo Librandi, Sep 06 2018

q[x_]:= x + 1;
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, 1, 40}];
Table[Coefficient[Part[t, n], x, 0], {n, 1, 40}] (* A192235 *)
Table[Coefficient[Part[t, n], x, 1], {n, 1, 40}] (* A192236 *)
Table[Coefficient[Part[t, n]/2, x, 1], {n, 1, 40}] (* A192237 *)
(* by Peter J. C. Moses, Jun 25 2011 *)
LinearRecurrence[{2,2,2,-1}, {0,0,0,1}, 40] (* Vincenzo Librandi, Sep 06 2018 *)

concat(vector(3), Vec(x^3/(1-2*x-2*x^2-2*x^3+x^4) + O(x^40))) \\ Colin Barker, Sep 06 2018

(x^3/(1-2*x-2*x^2-2*x^3+x^4)).series(x, 40).coefficients(x, sparse=False) # G. C. Greubel, Jul 30 2019

G.f.: x^3 / (1 - 2*x - 2*x^2 - 2*x^3 + x^4). - Colin Barker, Sep 12 2012 and Sep 06 2018

Entry revised (with new offset and initial terms) by N. J. A. Sloane, Sep 03 2018

A192243 0-sequence of reduction of Lucas sequence by x^2 -> x+1.

Original entry on oeis.org

1, 1, 5, 12, 34, 88, 233, 609, 1597, 4180, 10946, 28656, 75025, 196417, 514229, 1346268, 3524578, 9227464, 24157817, 63245985, 165580141, 433494436, 1134903170, 2971215072, 7778742049, 20365011073, 53316291173, 139583862444, 365435296162

Offset: 1

Clark Kimberling, Jun 26 2011

nonn

See A192232 for definition of "k-sequence of reduction of [sequence] by [substitution]".

Number of rooted ordered trees with n non-root nodes such that successive branch heights are weakly decreasing; examples are given in the Arndt link. - Joerg Arndt, Aug 27 2014

			The Lucas sequence provides coefficients for the power series 1+3x+4x^2+7x^3+..., whose partial sums are polynomials to which we apply reduction by x^2 -> x+1 as introduced at A192232:
1 -> 1
1+3x -> 1+3x
1+3x+4x^2 -> 1+3x+4(x+1)= 5+7x
1+3x+4x^2+7x^2 -> 12+21x..., so that
0-sequence=(1,1,5,12,...), 1-sequence=(0,3,7,21,...).

G. C. Greubel, Table of n, a(n) for n = 1..1000
Joerg Arndt, trees described in comment for 1<=n<=5
Index entries for linear recurrences with constant coefficients, signature (3,0,-3,1).

Cf. A192232, A192068.

I:=[1, 1, 5, 12]; [n le 4 select I[n] else 3*Self(n-1) - 3*Self(n-3) + Self(n-4): n in [1..30]]; // G. C. Greubel, Dec 21 2017

c[n_] := LucasL[n]; Table[c[n], {n, 1, 15}]; q[x_] := x + 1; p[0, x_] :=
1; p[n_, x_] := p[n - 1, x] + (x^n)*c[n + 1]; 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] &, p[n, x]]]], {n, 0, 50}]
u = Table[Coefficient[Part[t, n], x, 0], {n, 1, 50}] (* A192243 *)
Table[Coefficient[Part[t, n], x, 1], {n, 1, 50}] (* A192068 *)
(* Peter J. C. Moses, Jun 26 2011 *)
Table[SeriesCoefficient[x*(1 - 2*x + 2*x^2)/(1 - 3*x + 3*x^3 - x^4), {x, 0, n}], {n, 1, 50}]
LinearRecurrence[{3,0,-3,1}, {1,1,5,12}, 30] (* G. C. Greubel, Dec 21 2017 *)
Table[If[EvenQ[n],Fibonacci[2*n-1]-1, Fibonacci[2*n-1]], {n,1,20}] (* Rigoberto Florez, Aug 29 2019 *)

x='x+O('x^30); Vec(x*(1-2*x+2*x^2)/(1-3*x+3*x^3-x^4)) \\ G. C. Greubel, Dec 21 2017

From Colin Barker, Feb 08 2012: (Start)
G.f.: x*(1-2*x+2*x^2)/(1-3*x+3*x^3-x^4).
a(n) = 3*a(n-1) - 3*a(n-3) + a(n-4).
(End)
a(n) = (-1)*(2^(-1-n)*(5*((-2)^n+2^n) + (-5+sqrt(5))*(3+sqrt(5))^n - (3-sqrt(5))^n*(5 + sqrt(5)))) / 5. - Colin Barker, Dec 22 2017
a(n) = F(2n-1)-1 if n is even and F(2n-1) if n is odd, where F(n) is the n-th Fibonacci number. - Rigoberto Florez, Aug 29 2019
E.g.f.: - cosh(x) + (1/5)*(cosh(3*x/2) + sinh(3*x/2))*(5*cosh(sqrt(5)*x/2) - sqrt(5)*sinh(sqrt(5)*x/2)). - Stefano Spezia, Aug 30 2019

A192352 Constant term of the reduction of the polynomial p(n,x)=(1/2)((x+1)^n+(x-1)^n) by x^2->x+1.

Original entry on oeis.org

1, 0, 2, 1, 9, 13, 51, 106, 322, 771, 2135, 5401, 14445, 37324, 98514, 256621, 673933, 1760997, 4615823, 12075526, 31628466, 82781215, 216761547, 567428401, 1485645049, 3889310328, 10182603746, 26657986681, 69792188337, 182717232061

Offset: 1

Clark Kimberling, Jun 29 2011

nonn

1

			For an introduction to reductions of polynomials by substitutions such as x^2->x+1, see A192232.
The first six polynomials p(n,x) and their reductions are as follows:
p(0,x)=1 -> 1
p(1,x)=x -> x
p(2,x)=1+x^2 -> 2+x
p(3,x)=3x+x^3 -> 1+5x
p(4,x)=1+6x^2+x^4 -> 9+9x
p(5,x)=5x+10x^3+x^5 -> 13+30x.
From these, we read
A192352=(1,0,2,1,9,13...) and A049602=(0,1,1,5,9,30...).

Cf. A192232, A049602.

q[x_] := x + 1; d = 1;
p[n_, x_] := ((x + d)^n + (x - d)^n )/
  2 (* similar to polynomials defined at A161516 *)
Table[Expand[p[n, x]], {n, 0, 6}]
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] &, p[n, x]]]], {n, 0, 30}]
Table[Coefficient[Part[t, n], x, 0], {n, 1, 30}]
  (* A192352 *)
Table[Coefficient[Part[t, n], x, 1], {n, 1, 30}]
  (* A049602 *)

Empirical G.f.: -x*(x^3-x^2-2*x+1)/((x^2-3*x+1)*(x^2-x-1)). [Colin Barker, Sep 11 2012]

A192457 Constant term of the reduction by x^2->x+2 of the polynomial p(n,x) defined below in Comments.

Original entry on oeis.org

0, 2, 14, 118, 1210, 14730, 208110, 3350550, 60580170, 1215657450, 26813382750, 644830644150, 16793095369050, 470839138619850, 14140985865756750, 452938463797569750, 15412288335824630250, 555226177657611710250, 21111260070730770690750

Offset: 0

Clark Kimberling, Jul 01 2011

nonn

The polynomial p(n,x) is defined by recursively by p(n,x)=(x+2n)*p(n-1,x) with p[0,x]=x. For an introduction to reductions of polynomials by substitutions such as x^2->x+2, see A192232.

			The first four polynomials p(n,x) and their reductions are as follows:
p(0,x)=x -> x
p(1,x)=x(2+x) -> 2+3x
p(2,x)=x(2+x)(4+x) -> 14+17x
p(3,x)=x(2+x)(4+x)(6+x) -> 118+133x.
From these, read
A192457=(0,2,14,118,...) and A192459=(1,3,17,133,...)

Cf. A192232, A192459, A192458.

q[x_] := x + 2; p[0, x_] := x;
p[n_, x_] := (x + 2 n)*p[n - 1, x] /; n > 0
Table[Simplify[p[n, x]], {n, 0, 5}]
reductionRules = {x^y_?EvenQ -> q[x]^(y/2),
   x^y_?OddQ -> x q[x]^((y - 1)/2)};
t = Table[FixedPoint[Expand[#1 /. reductionRules] &, p[n, x]], {n, 0, 30}]
Table[Coefficient[Part[t, n], x, 0], {n, 1, 16}]  (* A192457 *)
Table[Coefficient[Part[t, n]/2, x, 0], {n, 1, 16}]  (* A192458 *)
Table[Coefficient[Part[t, n], x, 1], {n, 1, 16}]  (* A192459 *)

a(n) = (2/3)*(2^n(n+1)! - (2n-1)!!). - Vaclav Potocek, Feb 04 2016

A192472 Constant term of the reduction by x^2->x+1 of the polynomial p(n,x)=1+x^n+x^(2n+2).

Original entry on oeis.org

3, 7, 15, 37, 93, 239, 619, 1611, 4203, 10981, 28713, 75115, 196563, 514463, 1346647, 3525189, 9228453, 24159415, 63248571, 165584323, 433501203, 1134914117, 2971232785, 7778770707, 20365057443, 53316366199, 139583983839, 365435492581

Offset: 1

Clark Kimberling, Jul 01 2011

nonn

For an introduction to reductions of polynomials by substitutions such as x^2->x+1, see A192232.

			The first four polynomials p(n,x) and their reductions are as follows:
p(1,x)=1+x+x^4 -> 3+4x
p(2,x)=1+x^2+x^6 -> 7+9x
p(3,x)=1+x^3+x^8 -> 15+23x
p(4,x)=1+x^4+x^10 -> 37+58x.
From these, read
A192472=(3,7,15,37,...) and A192473=(4,9,23,58,...)

Cf. A192232, A192473.

Remove["Global`*"];
q[x_] := x + 1; p[n_, x_] := 1 + x^n + x^(2 n+2);
Table[Simplify[p[n, x]], {n, 1, 5}]
reductionRules = {x^y_?EvenQ -> q[x]^(y/2),
   x^y_?OddQ -> x q[x]^((y - 1)/2)};
t = Table[FixedPoint[Expand[#1 /. reductionRules] &, p[n, x]], {n, 1, 30}]
Table[Coefficient[Part[t, n], x, 0], {n, 1, 30}]
(* A192472 *)
Table[Coefficient[Part[t, n], x, 1], {n, 1, 30}]
(* A192473 *)

Empirical G.f.: -x*(3*x^4-8*x^3-x^2+8*x-3)/((x-1)*(x^2-3*x+1)*(x^2+x-1)). [Colin Barker, Nov 12 2012]

A192746 Constant term of the reduction by x^2 -> x+1 of the polynomial p(n,x) defined below in Comments.

Original entry on oeis.org

1, 5, 9, 17, 29, 49, 81, 133, 217, 353, 573, 929, 1505, 2437, 3945, 6385, 10333, 16721, 27057, 43781, 70841, 114625, 185469, 300097, 485569, 785669, 1271241, 2056913, 3328157, 5385073, 8713233, 14098309, 22811545, 36909857, 59721405, 96631265

Offset: 0

Clark Kimberling, Jul 09 2011

nonn

The titular polynomial is defined recursively by p(n,x)=x*(n-1,x)+3n+1 for n>0, where p(0,x)=1. For discussions of polynomial reduction, see A192232 and A192744.

G. C. Greubel, 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).

Cf. A000045, A192232, A192744.

List([0..30], n-> 4*Fibonacci(n+2)-3); # G. C. Greubel, Jul 24 2019

[4*Fibonacci(n+2)-3: n in [0..30]]; // G. C. Greubel, Jul 24 2019

(* First program *)
q = x^2; s = x + 1; z = 40;
p[0, n_]:= 1; p[n_, x_]:= x*p[n-1, x] +3n +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}] (* A192746 *)
u2 = Table[Coefficient[Part[t, n], x, 1], {n, 1, z}] (* A192747 *) (* Clark Kimberling, Jul 09 2011 *)
(* Additional programs *)
a[0]=1;a[1]=5;a[n_]:=a[n]=a[n-1]+a[n-2]+3;Table[a[n],{n,0,36}] (* Gerry Martens, Jul 04 2015 *)
4*Fibonacci[Range[0,40]+2]-3 (* G. C. Greubel, Jul 24 2019 *)

vector(30, n, n--; 4*fibonacci(n+2)-3) \\ G. C. Greubel, Jul 24 2019

[4*fibonacci(n+2)-3 for n in (0..30)] # G. C. Greubel, Jul 24 2019

G.f.: (1+3*x-x^2)/((1-x)*(1-x-x^2)), so the first differences are (essentially) A022087. - R. J. Mathar, May 04 2014

a(n) = 4*Fibonacci(n+2)-3. - Gerry Martens, Jul 04 2015

cp's OEIS Frontend

A265398 Perform one x^2 -> x+1 reduction for the polynomial with nonnegative integer coefficients that is encoded in the prime factorization of n.

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A265399 Repeatedly perform x^2 -> x+1 reduction for polynomial (with nonnegative integer coefficients) encoded in prime factorization of n, until the polynomial is at most degree 1.

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A192234 a(n) = 2*(a(n-1) + a(n-2) + a(n-3)) - a(n-4) for n >= 4, with initial terms 0,1,0,1.

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A192235 Constant term of the reduction of the n-th 2nd-kind Chebyshev polynomial by x^2 -> x+1.

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A192237 a(n) = 2*(a(n-1) + a(n-2) + a(n-3)) - a(n-4) for n >= 4, with initial terms 0,0,0,1.

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A192243 0-sequence of reduction of Lucas sequence by x^2 -> x+1.

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