cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

User: Ding Hao

Ding Hao's wiki page.

Ding Hao has authored 2 sequences.

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

Views

Author

Greg Dresden and Ding Hao, Aug 12 2022

Keywords

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.

Examples

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

Links

Crossrefs

Cf. A192380, A158934 (-c2).

Programs

  • Mathematica
    Table[Sum[(Sec[k Pi/20]/Sqrt[2] - 1)^n, {k, {1, 7, 9, 17}}], {n, 0, 30}] // Round
  • PARI
    polsym(x^4 - 2*x^3 - 6*x^2 + 2*x + 1, 22) \\ Joerg Arndt, Aug 14 2022

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).

A356463 Sum of powers of roots of x^3 - 4*x^2 + x + 1.

Original entry on oeis.org

3, 4, 14, 49, 178, 649, 2369, 8649, 31578, 115294, 420949, 1536924, 5611453, 20487939, 74803379, 273114124, 997165178, 3640743209, 13292693534, 48532865749, 177198026253, 646966545729, 2362135290914
Offset: 0

Views

Author

Greg Dresden and Ding Hao, Aug 08 2022

Keywords

Comments

The three roots of x^3 - 4*x^2 + x + 1 are c1 = 1 + 2*cos(Pi/13) + 2*cos(5*Pi/13), c2 = 1 + 2*cos(3*Pi/13) + 2*cos(11*Pi/13), and c3 = 1 + 2*cos(7*Pi/13) + 2*cos(9*Pi/13), so our entries are a(n) = c1^n + c2^n + c3^n.

Links

Crossrefs

Cf. A052941.

Programs

  • Mathematica
    LinearRecurrence[{4, -1, -1}, {3, 4, 14},30]
  • PARI
    a(n) = ([0,1,0;0,0,1;-1,-1,4]^n * [3;4;14])[1,1] \\ Jianing Song, Aug 09 2022
  • PARI
    polsym(x^3 - 4*x^2 + x + 1, 33) \\ Joerg Arndt, Sep 13 2022

Formula

G.f.: (3 - 8*x + x^2)/(1 - 4*x + x^2 + x^3).
a(n) = 4*a(n-1) - a(n-2) - a(n-3).
a(n) = 3*b(n-1) + 5*b(n-2) + b(n-3) for b(n) = A052941(n).
a(n) = round(c1^n), c1 per the comment, n >= 3. - Bill McEachen, Sep 12 2022

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