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 1-3 of 3 results.

A274664 Sum of n-th powers of the roots of x^3 + 11*x^2 - 4*x - 1.

Original entry on oeis.org

3, -11, 129, -1460, 165655, -187926, 2131986, -24186985, 274396853, -3112981337, 35316195134, -400655674969, 4545364223858, -51566312967180, 585010243859443, -6636832570098735, 75293632933556677, -854192282305658944, 9690652804526376357, -109938656346079219026, 1247233638742671255770
Offset: 0

Views

Author

Kai Wang, Jul 01 2016

Keywords

Comments

This is the other half for A274663.
a(n) is x1^n + x2^n + x3^n, where x1, x2, x3 are the roots of the polynomial x^3 + 11*x^2 - 3*x - 1.
x1 = (cos(2*Pi/7)*cos(4*Pi/7))/(cos(Pi/7))^2,
x2 = -(cos(4*Pi/7)*cos(Pi/7))/(cos(2*Pi/7))^2,
x3 = -(cos(Pi/7) *cos(2*Pi/7))(cos(4*Pi/7))^2.

Links

Crossrefs

Cf. A274663.

Programs

  • PARI
    Vec((3+22*x-4*x^2+149090*x^4+1639990*x^5-596360*x^6-149090*x^7) / (1+11*x-4*x^2-x^3) + O(x^20)) \\ Colin Barker, Jul 03 2016
  • PARI
    first(n)=my(x='x); polsym(x^3+11*x^2-4*x-1,n) \\ Charles R Greathouse IV, Jul 10 2016

Formula

a(0) = 3, a(1) = -11, a(2) = 129; thereafter a(n) = -11*a(n-1) + 4*a(n-2) + a(n-3).
a(n) = ((cos(2*Pi/7)*cos(4*Pi/7))/(cos(Pi/7))^2)^n +(-(cos(4*Pi/7)*cos(Pi/7))/(cos(2*Pi/7))^2)^n +(-(cos(Pi/7)*cos(2*Pi/7))/(cos(4*Pi/7))^2)^n.
G.f.: (3+22*x-4*x^2+149090*x^4+1639990*x^5-596360*x^6-149090*x^7) / (1+11*x-4*x^2-x^3). - Colin Barker, Jul 03 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

A322460 Sum of n-th powers of the roots of x^3 + 95*x^2 - 88*x - 1.

Original entry on oeis.org

3, -95, 9201, -882452, 84642533, -8118687210, 778722945402, -74693039645137, 7164358266796181, -687186244111463849, 65913082025027484446, -6322208017501153044901, 606409425694567846432994, -58165183833442021851601272, 5579050171430096545235179411
Offset: 0

Views

Author

Kai Wang, Dec 09 2018

Keywords

Comments

Let A = cos(2*Pi/7), B = cos(4*Pi/7), C = cos(8*Pi/7).
In general, for integer h, k let
X = A^(h+k)/(B^h*C^k),
Y = B^(h+k)/(C^h*A^k),
Z = C^(h+k)/(A^h*B^k).
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.
This sequence has (h,k) = (1,3).

Links

Crossrefs

Similar sequences with (h,k) values: A215076 (0,1), A274220 (1,0), A274663 (1,1), A248417 (1,2), A215560 (2,1).

Programs

  • Maple
    seq(coeff(series((3+190*x-88*x^2)/(1+95*x-88*x^2-x^3),x,n+1), x, n), n = 0 .. 20); # Muniru A Asiru, Dec 11 2018
  • Mathematica
    LinearRecurrence[{-95, 88, 1}, {3, -95, 9201}, 50] (* Amiram Eldar, Dec 09 2018 *)
  • PARI
    Vec((3 + 190*x - 88*x^2) / (1 + 95*x - 88*x^2 - x^3) + O(x^15)) \\ Colin Barker, Dec 09 2018
  • PARI
    polsym(x^3 + 95*x^2 - 88*x - 1, 25)  \\ Joerg Arndt, Dec 17 2018

Formula

a(n) = (A^4/(B*C^3))^n + (B^4/(C*A^3))^n + (C^4/(A*B^3))^n.
a(n) = -95*a(n-1) + 88*a(n-2) + a(n-3) for n>2.
G.f.: (3 + 190*x - 88*x^2) / (1 + 95*x - 88*x^2 - x^3). - Colin Barker, Dec 09 2018
Showing 1-3 of 3 results.

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