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

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

Offset

0,1

Comments

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.

Links

Table of n, a(n) for n=0..22.

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

Formula

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).

Example

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

Mathematica

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

Prog

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

Crossrefs

Cf. A192380, A158934 (-c2).

Sequence in context: A264195 A182872 A137393 * A122749 A189741 A381730

Adjacent sequences: A356566 A356567 A356568 * A356570 A356571 A356572

Keyword

nonn,easy

Author

Greg Dresden and Ding Hao, Aug 12 2022

Status

approved