A192385 -id:A192385 - cp's OEIS frontend

A192383 Constant term of the reduction by x^2 -> x+1 of the polynomial p(n,x) defined below in Comments.

1, 0, 6, 8, 60, 160, 744, 2496, 10064, 36480, 140512, 522624, 1983168, 7439360, 28091520, 105674752, 398391552, 1500057600, 5652182528, 21288560640, 80200784896, 302101094400, 1138045495296, 4286942363648, 16149041172480, 60833034895360

Offset: 1

Clark Kimberling, Jun 30 2011

nonn

The polynomial p(n,x) is defined by ((x+d)^n - (x-d)^n)/(2*d), where d=sqrt(x+3). For an introduction to reductions of polynomials by substitutions such as x^2 -> x+1, see A192232.

			The first five polynomials p(n,x) and their reductions are as follows:
  p(0, x) = 1 -> 1
  p(1, x) =     2*x -> 2*x
  p(2, x) = 3 +   x +  3*x^2 -> 6 + 4*x
  p(3, x) =    12*x +  4*x^2 +  4*x^3 -> 8 + 24*x
  p(4, x) = 9 + 6*x + 31*x^2 + 10*x^3 + 5*x^4 -> 60 + 72*x.
From these, read A192383 = (1, 0, 6, 8, 60, ...) and A192384 = (0, 2, 4, 24, 72, ...).

G. C. Greubel, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, signature (2,8,-4,-4).

Cf. A083084, A192232.

Cf. A192384, A192385.

R:=PowerSeriesRing(Integers(), 41);
Coefficients(R!( x*(1-2*x-2*x^2)/(1-2*x-8*x^2+4*x^3+4*x^4) )) // G. C. Greubel, Jul 10 2023

q[x_]:= x+1; d= Sqrt[x+3];
p[n_, x_]:= ((x+d)^n - (x-d)^n)/(2*d); (* suggested by A162517 *)
Table[Expand[p[n, x]], {n,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, 1, 30}];
Table[Coefficient[Part[t, n], x, 0], {n,30}] (* A192383 *)
Table[Coefficient[Part[t, n], x, 1], {n,30}] (* A192384 *)
Table[Coefficient[Part[t, n]/2, x, 1], {n,30}] (* A192385 *)
LinearRecurrence[{2,8,-4,-4}, {1,0,6,8}, 40] (* G. C. Greubel, Jul 10 2023 *)

@CachedFunction
def a(n): # a = A192383
    if (n<5): return (0,1,0,6,8)[n]
    else: return 2*a(n-1) +8*a(n-2) -4*a(n-3) -4*a(n-4)
[a(n) for n in range(1,41)] # G. C. Greubel, Jul 10 2023

From Colin Barker, May 11 2014: (Start)
a(n) = 2*a(n-1) + 8*a(n-2) - 4*a(n-3) - 4*a(n-4).
G.f.: x*(1-2*x-2*x^2)/(1-2*x-8*x^2+4*x^3+4*x^4). (End)
From G. C. Greubel, Jul 10 2023: (Start)
T(n, k) = [x^k] ((x+sqrt(x+3))^n - (x-sqrt(x+3))^n)/(2*sqrt(x+3)).
a(n) = Sum_{k=0..n-1} T(n, k)*Fibonacci(k-1). (End)

A192384 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, 2, 4, 24, 72, 312, 1088, 4288, 15744, 60192, 224832, 851072, 3197056, 12062592, 45398016, 171104256, 644354048, 2427699712, 9144222720, 34448209920, 129761986560, 488821962752, 1841370087424, 6936475090944, 26129575084032

Offset: 1

Clark Kimberling, Jun 30 2011

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

The polynomial p(n,x) is defined by ((x+d)^n - (x-d)^n)/(2*d), where d = sqrt(x+3). For an introduction to reductions of polynomials by substitutions such as x^2 -> x+1, see A192232.

			The first five polynomials p(n,x) and their reductions are as follows:
  p(0, x) = 1 -> 1
  p(1, x) =     2*x -> 2*x
  p(2, x) = 3 +   x +  3*x^2 -> 6 + 4*x
  p(3, x) =    12*x +  4*x^2 +  4*x^3 -> 8 + 24*x
  p(4, x) = 9 + 6*x + 31*x^2 + 10*x^3 + 5*x^4 -> 60 + 72*x.
From these, read A192383 = (1, 0, 6, 8, 60, ...) and A192384 = (0, 2, 4, 24, 72, ...).

G. C. Greubel, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, signature (2,8,-4,-4).

Cf. A192232, A192383, A192385.

R:=PowerSeriesRing(Integers(), 41);
[0] cat Coefficients(R!( 2*x^2/(1-2*x-8*x^2+4*x^3+4*x^4) )) // G. C. Greubel, Jul 10 2023

(See A192383.)
LinearRecurrence[{2,8,-4,-4}, {0,2,4,24}, 40] (* G. C. Greubel, Jul 10 2023 *)

@CachedFunction
def a(n): # a = A192384
    if (n<5): return (0,0,2,4,24)[n]
    else: return 2*a(n-1) +8*a(n-2) -4*a(n-3) -4*a(n-4)
[a(n) for n in range(1,41)] # G. C. Greubel, Jul 10 2023

From Colin Barker, Dec 09 2012: (Start)
a(n) = 2*a(n-1) + 8*a(n-2) - 4*a(n-3) - 4*a(n-4).
G.f.: 2*x^2/(1-2*x-8*x^2+4*x^3+4*x^4). (End)
From G. C. Greubel, Jul 10 2023: (Start)
T(n, k) = [x^k] ((x+sqrt(x+3))^n - (x-sqrt(x+3))^n)/(2*sqrt(x+3)).
a(n) = Sum_{k=0..n-1} T(n, k)*Fibonacci(k). (End)

cp's OEIS Frontend

A192383 Constant term of the reduction by x^2 -> x+1 of the polynomial p(n,x) defined below in Comments.

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