A192380 -id:A192380 - cp's OEIS frontend

A192381 (A192380)/2.

0, 1, 2, 10, 30, 115, 388, 1396, 4860, 17205, 60390, 212894, 748858, 2637095, 9281160, 32674280, 115012472, 404871209, 1425187530, 5016903090, 17660176470, 62166525211, 218835115532, 770332126300, 2711681718900, 9545519439325, 33601629823918

Offset: 1

Clark Kimberling, Jun 29 2011

nonn

(See A192380.)

			(See A192380.)

Cf. A192232, A192379, A192380.

Mathematica
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(See A192379.)
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A192379 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, 0, 5, 8, 45, 128, 505, 1680, 6089, 21120, 74909, 262680, 926485, 3258112, 11474865, 40382752, 142171985, 500432640, 1761656821, 6201182760, 21829269181, 76841888640, 270495370025, 952182350768, 3351823875225, 11798909226368

Offset: 1

Clark Kimberling, Jun 29 2011

nonn

The polynomial p(n,x) is defined by ((x+d)^n-(x-d)^n)/(2d), where d=sqrt(x+2). 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)=2x -> 2x
p(2,x)=2+x+3x^2 -> 5+4x
p(3,x)=8x+4x^2+4x^3 -> 8+20x
p(4,x)=4+4x+21x^2+10x^3+5x^4 -> 45+60x.
From these, read A192379=(1,0,5,8,45,...) and A192380=(0,2,4,20,60,...).

Cf. A192232, A192380, A192381.

q[x_] := x + 1; d = Sqrt[x + 2];
p[n_, x_] := ((x + d)^n - (x - d)^n )/(2 d)   (* Cf. A162517 *)
Table[Expand[p[n, x]], {n, 1, 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, 1, 30}]
Table[Coefficient[Part[t, n], x, 0], {n, 1, 30}]   (* A192379 *)
Table[Coefficient[Part[t, n], x, 1], {n, 1, 30}]   (* A192380 *)
Table[Coefficient[Part[t, n]/2, x, 1], {n, 1, 30}]   (* A192381 *)

Conjecture: a(n) = 2*a(n-1)+6*a(n-2)-2*a(n-3)-a(n-4). G.f.: -x*(x^2+2*x-1) / (x^4+2*x^3-6*x^2-2*x+1). - Colin Barker, May 11 2014

A356569 Sums of powers of roots of x^4 - 2x^3 - 6x^2 + 2*x + 1.

Original entry on oeis.org

4, 2, 16, 38, 164, 522, 1936, 6638, 23684, 82802, 292496, 1027798, 3621284, 12741562, 44862736, 157904478, 555880964, 1956721762, 6888057616, 24246779398, 85352580004, 300452999402, 1057639862416

Offset: 0

Greg Dresden and Ding Hao, Aug 12 2022

The four roots of x^4 - 2*x^3 - 6*x^2 + 2*x + 1, in order from smallest to largest, are c1 = sec(17*Pi/20)/sqrt(2) - 1, c2 = sec(Pi/20)/sqrt(2) - 1 = -A158934, c3 = sec(7*Pi/20)/sqrt(2) - 1, and c4 = sec(9*Pi/20)/sqrt(2) - 1.

			a(3) = (-1.7936045...)^3 + (-0.28407904...)^3 + (0.55753652...)^3 + (3.5201470...)^3 = 38, as expected.

Index entries for linear recurrences with constant coefficients, signature (2,6,-2,-1).

Cf. A192380, A158934 (-c2).

Table[Sum[(Sec[k Pi/20]/Sqrt[2] - 1)^n, {k, {1, 7, 9, 17}}], {n, 0, 30}] // Round

polsym(x^4 - 2*x^3 - 6*x^2 + 2*x + 1, 22) \\ Joerg Arndt, Aug 14 2022

a(n) = 2*a(n-1) + 6*a(n-2) - 2*a(n-3) - a(n-4).
G.f.: 2*(2-3*x-6*x^2+x^3)/(1-2*x-6*x^2+2*x^3+x^4).
a(n) = b(n+1) + 6*b(n) - 3*b(n-1) - 2*b(n-2) for b(n) = A192380(n).

cp's OEIS Frontend

A192381 (A192380)/2.

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A192379 Constant term of the reduction by x^2->x+1 of the polynomial p(n,x) defined below in Comments.

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A356569 Sums of powers of roots of x^4 - 2x^3 - 6x^2 + 2*x + 1.

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cp's OEIS Frontend

A192381 (A192380)/2.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica

A192379 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

A356569 Sums of powers of roots of x^4 - 2*x^3 - 6*x^2 + 2*x + 1.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Formula

A356569 Sums of powers of roots of x^4 - 2x^3 - 6x^2 + 2*x + 1.