A274220 -id:A274220 - 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

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

A287396 a(n) = (7(csc(2Pi/7))^2)^n + (7(csc(4Pi/7))^2)^n + (7(csc(8Pi/7))^2)^n.

Original entry on oeis.org

3, 56, 1568, 53312, 1931776, 71300096, 2645479424, 98305622016, 3654656065536, 135885355483136, 5052615982317568, 187873377732526080, 6985794697679601664, 259756778648305139712, 9658687473893481906176, 359144636249686988029952, 13354285908291066433372160

Offset: 0

Kai Wang, May 24 2017

Colin Barker, Table of n, a(n) for n = 0..600
Roman Witula, Ramanujan Type Trigonometric Formulas: The General Form for the Argument 2Pi/7, J. Integer Seq., 12 (2009), Article 09.8.5.
Roman Witula and Damian Slota, New Ramanujan-Type Formulas and Quasi-Fibonacci Numbers of Order 7, Journal of Integer Sequences, Vol. 10 (2007), Article 07.5.6
Roman Witula and Damian Slota, Quasi-Fibonacci Numbers of Order 11, Journal of Integer Sequences, Vol. 10 (2007), Article 07.8.5
Roman Witula, Damian Slota and Adam Warzynski, Quasi-Fibonacci Numbers of the Seventh Order, J. Integer Seq., 9 (2006), Article 06.4.3.
Index entries for linear recurrences with constant coefficients, signature (56,-784,3136).

Cf. A033304, A094648, A274220, A215076, A274075, A274032, A275195, A215575, A274975.

LinearRecurrence[{56,-784,3136},{3,56,1568},30] (* Harvey P. Dale, Aug 08 2017 *)

Vec((3 - 28*x)*(1 - 28*x) / (1 - 56*x + 784*x^2 - 3136*x^3) + O(x^30)) \\ Colin Barker, May 25 2017

polsym(x^3 - 56*x^2 + 784* x - 3136, 20) \\ Joerg Arndt, May 26 2017

a(n) = x1^n + x2^n + x3^n, where x1, x2, x3 are the roots of x^3 - 56*x^2 + 784* x - 3136, x1 = 7*(csc(2*Pi/7))^2, x2 = 7*(csc(4*Pi/7))^2, x3 = 7*(csc(8*Pi/7))^2.
a(n) = 56*a(n-1) - 784*a(n-2) + 3136*a(n-3) for n>2, a(0) = 3, a(1) = 56, a(2) = 1568.
G.f.: (3 - 28*x)*(1 - 28*x) / (1 - 56*x + 784*x^2 - 3136*x^3). - Colin Barker, May 25 2017

A287405 a(n) = (7(cot(1Pi/7))^2)^n + (7(cot(2Pi/7))^2)^n + (7(cot(4Pi/7))^2)^n.

Original entry on oeis.org

3, 35, 931, 27587, 830403, 25054435, 756187747, 22824258947, 688917131651, 20793986742179, 627637106311971, 18944339609269571, 571808137046942019, 17259221092289630307, 520945214725090792931, 15723995613526902256387, 474606601742375424297731

Offset: 0

Kai Wang, May 24 2017

a(n) = x1^n + x2^n + x3^n, where x1, x2, x3 are the roots of x^3 - 35*x^2 + 147*x - 49, x1 = 7*(cot(1*Pi/7))^2, x2 = 7*(cot(2*Pi/7))^2, x3 = 7*(cot(4*Pi/7))^2.

Colin Barker, Table of n, a(n) for n = 0..650
Roman Witula, Ramanujan Type Trigonometric Formulas: The General Form for the Argument 2Pi/7, J. Integer Seq., 12 (2009), Article 09.8.5.
Roman Witula and Damian Slota, New Ramanujan-Type Formulas and Quasi-Fibonacci Numbers of Order 7, Journal of Integer Sequences, Vol. 10 (2007), Article 07.5.6.
Index entries for linear recurrences with constant coefficients, signature (35,-147,49).

Cf. A108716, A033304, A094648, A274220, A215076, A274075 A274032, A275195, A215575, A274975.

LinearRecurrence[{35,-147,49},{3,35,931},30] (* Harvey P. Dale, Mar 15 2018 *)

Vec((3 - 7*x)*(1 - 21*x) / (1 - 35*x + 147*x^2 - 49*x^3) + O(x^30)) \\ Colin Barker, May 26 2017

polsym(x^3 - 35*x^2 + 147*x - 49, 20) \\ Joerg Arndt, May 26 2017

a(n) = 35*a(n-1) - 147*a(n-2) + 49*a(n-3), a(0) = 3, a(1) = 35, a(2) = 931.
Bisection of A215575: a(n) = A215575(2*n).
G.f.: (3 - 7*x)*(1 - 21*x) / (1 - 35*x + 147*x^2 - 49*x^3). - Colin Barker, May 26 2017

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

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A287396 a(n) = (7*(csc(2*Pi/7))^2)^n + (7*(csc(4*Pi/7))^2)^n + (7*(csc(8*Pi/7))^2)^n.

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A287405 a(n) = (7*(cot(1*Pi/7))^2)^n + (7*(cot(2*Pi/7))^2)^n + (7*(cot(4*Pi/7))^2)^n.

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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.

A287396 a(n) = (7(csc(2Pi/7))^2)^n + (7(csc(4Pi/7))^2)^n + (7(csc(8Pi/7))^2)^n.

A287405 a(n) = (7(cot(1Pi/7))^2)^n + (7(cot(2Pi/7))^2)^n + (7(cot(4Pi/7))^2)^n.

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