|
|
A192428
|
|
Constant term of the reduction by x^2 -> x+1 of the polynomial p(n,x) defined below in Comments.
|
|
2
|
|
|
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
(list;
graph;
refs;
listen;
history;
text;
internal format)
|
|
|
OFFSET
|
0,3
|
|
COMMENTS
|
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.
|
|
LINKS
|
|
|
FORMULA
|
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
|
|
EXAMPLE
|
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, ...).
|
|
MATHEMATICA
|
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 *)
|
|
PROG
|
(Magma) R<x>:=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
(SageMath)
@CachedFunction
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)
|
|
CROSSREFS
|
|
|
KEYWORD
|
nonn
|
|
AUTHOR
|
|
|
STATUS
|
approved
|
|
|
|