A248417 -id:A248417 - cp's OEIS frontend

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

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

Offset: 0

Kai Wang, Jul 01 2016

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)).

Colin Barker, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (-4,11,1).

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

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)
```

Vec((3+8*x-11*x^2)/(1+4*x-11*x^2-x^3) + O(x^99)) \\ Altug Alkan, Jul 08 2016

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

A274592 Sum of n-th powers of the roots of x^3 -31* x^2 - 25*x - 1.

Original entry on oeis.org

3, 31, 1011, 32119, 1020995, 32454831, 1031656755, 32793751175, 1042430160131, 33136210400191, 1053316070160371, 33482245865136407, 1064315659783638083, 33831894915991351119, 1075430116136187973171, 34185195288781394584359, 1086660638750543922795523

Offset: 0

Kai Wang, Jun 29 2016

This is one side of a two sided sequence (see A248417).
a(n) is x1^n + x2^n + x3^n, where x1, x2, x3 are the roots of the polynomial
x^3 -31* x^2 - 25*x - 1.
x1 = (tan(Pi/7))^2/(tan(2*Pi/7)*tan(4*Pi/7)),
x2 = (tan(2*Pi/7))^2/(tan(4*Pi/7)*tan(Pi/7)),
x3 = (tan(4*Pi/7))^2/(tan(Pi/7)*tan(2*Pi/7)).

Colin Barker, Table of n, a(n) for n = 0..600
Index entries for linear recurrences with constant coefficients, signature (31,25,1).

Cf. A248417, A274032, A274075.

LinearRecurrence[{31,25,1},{3,31,1011},20] (* Harvey P. Dale, Feb 02 2022 *)

Vec((3-62*x-25*x^2)/(1-31*x-25*x^2-x^3) + O(x^20)) \\ Colin Barker, Jun 30 2016

polsym(x^3 -31* x^2 - 25*x - 1, 30) \\ Charles R Greathouse IV, Jul 20 2016

a(n) = ((tan(Pi/7))^2/(tan(2*Pi/7)*tan(4*Pi/7)))^n + ((tan(2*Pi/7))^2/(tan(4*Pi/7)*tan(Pi/7)))^n + ((tan(4*Pi/7))^2/(tan(Pi/7)*tan(2*Pi/7)))^n.
a(n) = 31*a(n-1) + 25*a(n-2) + a(n-3).
G.f.: (3-62*x-25*x^2) / (1-31*x-25*x^2-x^3). - Colin Barker, Jun 30 2016

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

Kai Wang, Oct 24 2018

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.

Colin Barker, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (-9,-20,1).

Cf. A248417, A274032, A274075.

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

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 *)

polsym(x^3 + 9*x^2 + 20*x - 1, 25) \\ Joerg Arndt, Oct 24 2018

Vec((3 + 18*x + 20*x^2) / (1 + 9*x + 20*x^2 - x^3) + O(x^30)) \\ Colin Barker, Dec 09 2018

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 + 95x^2 - 88x - 1.

Original entry on oeis.org

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

Offset: 0

Kai Wang, Dec 09 2018

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).

Colin Barker, Table of n, a(n) for n = 0..500
Index entries for linear recurrences with constant coefficients, signature (-95,88,1).

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

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

LinearRecurrence[{-95, 88, 1}, {3, -95, 9201}, 50] (* Amiram Eldar, Dec 09 2018 *)

Vec((3 + 190*x - 88*x^2) / (1 + 95*x - 88*x^2 - x^3) + O(x^15)) \\ Colin Barker, Dec 09 2018

polsym(x^3 + 95*x^2 - 88*x - 1, 25)  \\ Joerg Arndt, Dec 17 2018

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

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

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A274592 Sum of n-th powers of the roots of x^3 -31* x^2 - 25*x - 1.

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

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

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

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

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