A163762 -id:A163762 - cp's OEIS frontend

A192428 Constant term of the reduction by x^2 -> x+1 of the polynomial p(n,x) defined below in Comments.

1, 1, 5, 11, 57, 185, 829, 3067, 12801, 49633, 201413, 794747, 3190617, 12673529, 50672029, 201782923, 805529409, 3210794113, 12810136517, 51078991403, 203744818617, 812521585145, 3240726179389, 12924488375899, 51547405667265

Offset: 0

Clark Kimberling, Jun 30 2011

nonn

The polynomial p(n,x) is defined by ((x+d)^n + (x-d)^n)/2 + ((x+d)^n - (x-d)^n)/(2*d), where d = sqrt(x^2+4), as at A163762. For an introduction to reductions of polynomials by substitutions such as x^2 -> x+2, see A192232.

			The first five polynomials p(n,x) and their reductions are as follows:
  p(0,x) = 1 -> 1
  p(1,x) = 1 + x -> 1 + x
  p(2,x) = 4 + 3*x + x^2 -> 5 + 4*x
  p(3,x) = 4 + 13*x + 6*x^2 + x^3 -> 11 + 21*x
  p(4,x) = 16 + 24*x + 29*x^2 + 10*x^3 + x^4 -> 57 + 76*x.
From these, read a(n) = (1, 1, 5, 11, 57, 185, ...) and A192429 = (0, 1, 4, 21, 76, 329, ...).

G. C. Greubel, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (2,10,-6,-9).

Cf. A163762, A192232, A192429.

R:=PowerSeriesRing(Integers(), 40); Coefficients(R!( (1-x-7*x^2-3*x^3)/(1-2*x-10*x^2+6*x^3+9*x^4) )); // G. C. Greubel, Jul 13 2023

q[x_]:= x+1; d= Sqrt[x+4];
u[x_]:= x+d; v[x_]:= x-d;
p[n_, x_]:= (u[x]^n +v[x]^n)/2 + (u[x]^n -v[x]^n)/(2*d) (* A163762 *)
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[FixedPoint[Expand[#1 /. reductionRules] &, p[n, x]], {n, 0, 30}]
Table[Coefficient[Part[t, n], x, 0], {n,30}] (* A192428 *)
Table[Coefficient[Part[t, n], x, 1], {n,30}] (* A192429 *)
LinearRecurrence[{2,10,-6,-9}, {1,1,5,11}, 40] (* G. C. Greubel, Jul 13 2023 *)

@CachedFunction
def a(n): # a = A192428
    if (n<4): return (1, 1, 5, 11)[n]
    else: return 2*a(n-1) + 10*a(n-2) - 6*a(n-3) - 9*a(n-4)
[a(n) for n in range(41)] # G. C. Greubel, Jul 13 2023

From Colin Barker, May 12 2014: (Start)
a(n) = 2*a(n-1) + 10*a(n-2) - 6*a(n-3) - 9*a(n-4).
G.f.: (1-x-7*x^2-3*x^3)/(1-2*x-10*x^2+6*x^3+9*x^4). (End)
a(n) = Sum_{k=0..n} T(n,k)*Fibonacci(k-1), where T(n, k) = [x^k] ( ((x + sqrt(x+4))^n + (x - sqrt(x+4))^n)/2 + ((x + sqrt(x+4))^n - (x - sqrt(x+4))^n)/(2*sqrt(x+4)) ). - G. C. Greubel, Jul 13 2023

A192429 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, 21, 76, 329, 1256, 5157, 20216, 81505, 322924, 1293189, 5144644, 20550089, 81881168, 326756661, 1302722672, 5196774145, 20723304532, 82657204533, 329642305468, 1314745861769, 5243461810232, 20912613564549, 83404589311592

Offset: 0

Clark Kimberling, Jun 30 2011

nonn

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

			The first five polynomials p(n,x) and their reductions are as follows:
  p(0,x) = 1 -> 1
  p(1,x) = 1 + x -> 1 + x
  p(2,x) = 4 + 3*x + x^2 -> 5 + 4*x
  p(3,x) = 4 + 13*x + 6*x^2 + x^3 -> 11 + 21*x
  p(4,x) = 16 + 24*x + 29*x^2 + 10*x^3 + x^4 -> 57 + 76*x.
From these, read A192428 = (1, 1, 5, 11, 57, 185, ...) and a(n) = (0, 1, 4, 21, 76, 329, ...).

G. C. Greubel, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (2,10,-6,-9).

Cf. A000045, A163762, A192232, A192428.

R:=PowerSeriesRing(Integers(), 40); [0] cat Coefficients(R!( x*(1+2*x+3*x^2)/(1-2*x-10*x^2+6*x^3+9*x^4) )); // G. C. Greubel, Jul 13 2023

(See A192428.)
LinearRecurrence[{2,10,-6,-9}, {0,1,4,21}, 40] (* G. C. Greubel, Jul 13 2023 *)

@CachedFunction
def a(n): # a = A192429
    if (n<4): return (0,1,4,21)[n]
    else: return 2*a(n-1) + 10*a(n-2) - 6*a(n-3) - 9*a(n-4)
[a(n) for n in range(41)] # G. C. Greubel, Jul 13 2023

From Colin Barker, May 12 2014: (Start)
a(n) = 2*a(n-1) + 10*a(n-2) - 6*a(n-3) - 9*a(n-4).
G.f.: x*(1+2*x+3*x^2)/(1-2*x-10*x^2+6*x^3+9*x^4). (End)
a(n) = Sum_{k=0..n} T(n,k)*Fibonacci(k), where T(n, k) = [x^k] ( ((x + sqrt(x+4))^n + (x - sqrt(x+4))^n)/2 + ((x + sqrt(x+4))^n - (x - sqrt(x+4))^n)/(2*sqrt(x+4)) ). - G. C. Greubel, Jul 13 2023

A192430 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, 1, 3, 9, 33, 113, 403, 1409, 4977, 17489, 61619, 216809, 763377, 2686881, 9458787, 33295297, 117206177, 412579681, 1452347043, 5112464521, 17996645761, 63350804881, 223004208243, 785007489729, 2763341973393, 9727369663793

Offset: 0

Clark Kimberling, Jun 30 2011

nonn

The polynomial p(n,x) is defined by (u^n+v^n)//2)^n+(u^n-v^n)/(2d), where u=x+d, v=x-d, d=sqrt(x^2+2). For an introduction to reductions of polynomials by substitutions such as x^2->x+2, see A192232.

			The first five polynomials p(n,x) and their reductions are as follows:
p(0,x)=1 -> 1
p(1,x)=1+x -> 1+x
p(2,x)=2+3x+x^2 -> 3+4x
p(3,x)=2+7x+6x^2+x^3 -> 9+15x
p(4,x)=4+12x+17x^2+10x^3+x^4 -> 33+52x.
From these, read A192430=(1,1,3,9,33,...) and A192431=(0,1,4,15,52,...).

Cf. A192232, A192431.

q[x_] := x + 1; d = Sqrt[x + 2];
u[x_] := x + d; v[x_] := x - d;
p[n_, x_] := (u[x]^n + v[x]^n)/2 + (u[x]^n - v[x]^n)/(2 d) (* A163762 *)
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[FixedPoint[Expand[#1 /. reductionRules] &, p[n, x]], {n, 0, 30}]
Table[Coefficient[Part[t, n], x, 0], {n, 1, 30}] (* A192430 *)
Table[Coefficient[Part[t, n], x, 1], {n, 1, 30}] (* A192431 *)

Conjecture: a(n) = 2*a(n-1)+6*a(n-2)-2*a(n-3)-a(n-4). G.f.: -(x^3+5*x^2+x-1) / (x^4+2*x^3-6*x^2-2*x+1). - Colin Barker, May 12 2014

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A192428 Constant term of the reduction by x^2 -> x+1 of the polynomial p(n,x) defined below in Comments.

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A192429 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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Magma

Mathematica

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A192430 Constant term of the reduction by x^2->x+1 of the polynomial p(n,x) defined below in Comments.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica

Formula