A274075 -id:A274075 - cp's OEIS frontend

A274032 Sum of n-th powers of the roots of x^3 + 9*x^2 - x - 1.

3, -9, 83, -753, 6851, -62329, 567059, -5159009, 46935811, -427014249, 3884905043, -35344223825, 321555905219, -2925462465753, 26615373873171, -242142271419073, 2202970354179075, -20042260085157577, 182341168849178195, -1658909809373582257

Offset: 0

Kai Wang, Jun 07 2016

A Berndt-type sequence for tan(2*Pi/7).
a(n) is always an integer.
a(n) is x1^n + x2^n + x3^n, where x1, x2, x3 are the roots of the polynomial
x^3 + 9*x^2 - x - 1.
x1 = tan(Pi/7)/tan(2*Pi/7),
x2 = tan(2*Pi/7)/tan(4*Pi/7),
x3 = tan(4*Pi/7)/tan(Pi/7).
This is a two sided sequence. The other half is A274075. - Kai Wang, Aug 02 2016

Seiichi Manyama, Table of n, a(n) for n = 0..1000
B. C. Berndt, L.-C. Zhang, Ramanujan's identities for eta-functions, Math. Ann. 292 (1992), 561-573.
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 (-9,1,1).

Cf. A033304, A094648, A215076, A274075.

Vec((3+18*x-x^2)/(1+9*x-x^2-x^3) + O(x^30)) \\ Colin Barker, Jun 07 2016

polsym(x^3 + 9*x^2 - x - 1, 30) \\ Charles R Greathouse IV, Jul 20 2016

a(n) = (tan(Pi/7)/tan(2*Pi/7))^n + (-tan(2*Pi/7)/tan(3*Pi/7))^n + (-tan(3*Pi/7)/tan(Pi/7))^n.
From Colin Barker, Jun 07 2016: (Start)
a(n) = -9*a(n-1)+a(n-2)+a(n-3) for n>2.
G.f.: (3+18*x-x^2) / (1+9*x-x^2-x^3).
(End)

Edited by N. J. A. Sloane, Jun 07 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

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

A322461 Sum of n-th powers of the roots of x^3 + 8x^2 + 5x - 1.

Original entry on oeis.org

3, -8, 54, -389, 2834, -20673, 150825, -1100401, 8028410, -58574450, 427353149, -3117924532, 22748056061, -165967472679, 1210881576595, -8834467193304, 64455362190778, -470259679983109, 3430966161717678, -25031975531635101, 182630713764509309

Offset: 0

Kai Wang, Dec 09 2018

Let A = cos(2*Pi/7), B = cos(4*Pi/7), C = cos(8*Pi/7).
For integers h, k let
   X = 2*sqrt(7)*A^(h+k+1)/(B^h*C^k),
   Y = 2*sqrt(7)*B^(h+k+1)/(C^h*A^k),
   Z = 2*sqrt(7)*C^(h+k+1)/(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) = (0,1).

Colin Barker, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (-8,-5,1).

Similar sequences with (h,k) values: A094648 (0,0), A274075 (1,0).

LinearRecurrence[{-8, -5, 1}, {3, -8, 54}, 50] (* Amiram Eldar, Dec 09 2018 *)

Vec((3 + x)*(1 + 5*x) / (1 + 8*x + 5*x^2 - x^3) + O(x^25)) \\ Colin Barker, Dec 09 2018

polsym(x^3 + 8*x^2 + 5*x - 1, 25) \\ Joerg Arndt, Dec 17 2018

a(n) = (2*sqrt(7)*A^2/C)^n + (2*sqrt(7)*B^2/A)^n + (2*sqrt(7)*C^2/B)^n, where A = cos(2*Pi/7), B = cos(4*Pi/7), C = cos(8*Pi/7).
a(n) = -8*a(n-2) - 5*a(n-2) + a(n-3) for n > 2.
G.f.: (3 + x)*(1 + 5*x) / (1 + 8*x + 5*x^2 - x^3). - Colin Barker, Dec 09 2018

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A274032 Sum of n-th powers of the roots of x^3 + 9*x^2 - 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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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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A322461 Sum of n-th powers of the roots of x^3 + 8*x^2 + 5*x - 1.

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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.

A322461 Sum of n-th powers of the roots of x^3 + 8x^2 + 5x - 1.