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.

Showing 1-2 of 2 results.

A274663 Sum of n-th powers of the roots of x^3 + 4*x^2 - 11*x - 1.

Original entry on oeis.org

3, -4, 38, -193, 1186, -6829, 40169, -234609, 1373466, -8034394, 47011093, -275049240, 1609284589, -9415668903, 55089756851, -322322100748, 1885860059450, -11033893589177, 64557712909910, -377717821061137, 2209972232664381, -12930227249420121
Offset: 0

Views

Author

Kai Wang, Jul 01 2016

Keywords

Comments

This is half of a two sided sequences.
The other half is A274664. - Kai Wang, Aug 02 2016
a(n) is x1^n + x2^n + x3^n, where x1, x2, x3 are the roots of the polynomial x^3 + 4*x^2 - 11*x - 1.
x1 = (cos(Pi/7))^2/(cos(2*Pi/7)*cos(4*Pi/7)),
x2 = -(cos(2*Pi/7))^2/(cos(4*Pi/7)*cos(Pi/7)),
x3 = -(cos(4*Pi/7))^2/(cos(Pi/7) *cos(2*Pi/7)).

Links

Crossrefs

Cf. A215076, A248417, A274220, A274592, A274664.

Programs

  • Mathematica
    RecurrenceTable[{a[0] == 3, a[1] == -4, a[2] == 38, a[n] == -4 a[n - 1] + 11 a[n - 2] + a[n - 3]}, a, {n, 0, 20}] (* Michael De Vlieger, Jul 02 2016 *)
LinearRecurrence[{-4,11,1},{3,-4,38},30] (* Harvey P. Dale, Dec 28 2022 *)
  • PARI
    polsym(x^3 + 4*x^2 - 11*x - 1, 21)
  • PARI
    Vec((3+8*x-11*x^2)/(1+4*x-11*x^2-x^3) + O(x^99)) \\ Altug Alkan, Jul 08 2016

Formula

a(n) = ((cos(Pi/7))^2/(cos(2*Pi/7)*cos(4*Pi/7)))^n + (-(cos(2*Pi/7))^2/(cos(4*Pi/7)*cos(Pi/7)))^n + (-(cos(4*Pi/7))^2/(cos(Pi/7)*cos(2*Pi/7)))^n.
a(n) = -4*a(n-1) + 11*a(n-2) + a(n-3) for n>2.
G.f.: (3+8*x-11*x^2)/(1+4*x-11*x^2-x^3). - Wesley Ivan Hurt, Jul 02 2016
a(n) = (-1/8)^(-n)*cos(Pi/7)^(3*n) + (-8)^n*sin(Pi/14)^(3*n) +
8^n*sin(3*Pi/14)^(3*n). - Wesley Ivan Hurt, Jul 11 2016

A320918 Sum of n-th powers of the roots of x^3 + 9*x^2 + 20*x - 1.

Original entry on oeis.org

3, -9, 41, -186, 845, -3844, 17510, -79865, 364741, -1667859, 7636046, -35002493, 160633658, -738017016, 3394477491, -15629323441, 72036344133, -332346150886, 1534759151873, -7093873005004, 32817327856690, -151943731458257, 704053152985509, -3264786419847751
Offset: 0

Views

Author

Kai Wang, Oct 24 2018

Keywords

Comments

In general, for integer h, k let
X = (sin^(h+k)(2*Pi/7))/(sin^(h)(4*Pi/7)*sin^(k)(8*Pi/7)),
Y = (sin^(h+k)(4*Pi/7))/(sin^(h)(8*Pi/7)*sin^(k)(2*Pi/7)),
Z = (sin^(h+k)(8*Pi/7))/(sin^(h)(2*Pi/7)*sin^(k)(4*Pi/7)).
then X, Y, Z are the roots of a monic equation
t^3 + a*t^2 + b*t + c = 0
where a, b, c are integers and c = 1 or -1.
Then X^n + Y^n + Z^n, n = 0, 1, 2, ... is an integer sequence.
Instances of such sequences with (h,k) values:
(-3,0), (0,3), (3,-3): gives A274663;
(-3,3), (0,-3): give A274664;
(-2,0), (0,2), (2,-2): give A198636;
(-2,-3), (-1,-2), (2,-1), (3,-1): give A274032;
(-1,-1), (-1,2): give A215076;
(-1,0), (0,1), (1,-1): give A094648;
(-1,1), (0,-1), (1,0): give A274975;
(1,1), (-2,1), (1,-2): give A274220;
(1,2), (-3,1), (2,-3): give A274075;
(1,3): this sequence.

Links

Crossrefs

Cf. A248417, A274032, A274075.

Programs

  • Maple
    a := proc(n) option remember; if n < 3 then [3, -9, 41][n+1] else
-9*a(n-1) - 20*a(n-2) + a(n-3) fi end: seq(a(n), n=0..32); # Peter Luschny, Oct 25 2018
  • Mathematica
    CoefficientList[Series[(3 + 18*x + 20*x^2) / (1 + 9*x + 20*x^2 - x^3) , {x, 0, 50}], x] (* Amiram Eldar, Dec 09 2018 *)
LinearRecurrence[{-9,-20,1},{3,-9,41},30] (* Harvey P. Dale, Dec 10 2023 *)
  • PARI
    polsym(x^3 + 9*x^2 + 20*x - 1, 25) \\ Joerg Arndt, Oct 24 2018
  • PARI
    Vec((3 + 18*x + 20*x^2) / (1 + 9*x + 20*x^2 - x^3) + O(x^30)) \\ Colin Barker, Dec 09 2018

Formula

a(n) = ((sin^4(2*Pi/7))/(sin(4*Pi/7)*sin^3(8*Pi/7)))^n
+ ((sin^4(4*Pi/7))/(sin(8*Pi/7)*sin^3(2*Pi/7)))^n
+ ((sin^4(8*Pi/7))/(sin(2*Pi/7)*sin^3(4*Pi/7)))^n.
a(n) = -9*a(n-1) - 20*a(n-2) + a(n-3) for n>2.
G.f.: (3 + 18*x + 20*x^2) / (1 + 9*x + 20*x^2 - x^3). - Colin Barker, Dec 09 2018
Showing 1-2 of 2 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

Content is available under The OEIS End-User License Agreement.