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 1-7 of 7 results.

A274075 Sum of n-th powers of the roots of x^3 + x^2 - 9*x - 1.

Original entry on oeis.org

3, -1, 19, -25, 195, -401, 2131, -5545, 24323, -72097, 285459, -910009, 3407043, -11311665, 41065043, -139462985, 497736707, -1711838529, 6052005907, -20960815961, 73717030595, -256312368337, 898804827731, -3131899112169, 10964830193411, -38253117375201
Offset: 0

Views

Author

Kai Wang, Jun 09 2016

Keywords

Comments

a(n) is always an integer.
This is the other half of A274032.
a(n) is x1^n + x2^n + x3^n, where x1, x2, x3 are the roots of the polynomial
x^3 + x^2 - 9*x - 1.
x1 = tan(Pi/7)/tan(4*Pi/7),
x2 = tan(4*Pi/7)/tan(2*Pi/7),
x3 = tan(2*Pi/7)/tan(Pi/7).

Links

Crossrefs

Cf. A033304, A094648, A215076, A274032.

Programs

  • Mathematica
    FullSimplify[Table[(Tan[Pi/7]/Tan[4*Pi/7])^n + (Tan[4*Pi/7]/Tan[2*Pi/7])^n + (Tan[2*Pi/7]/Tan[Pi/7])^n, {n, 0, 12}]] (* Wesley Ivan Hurt, Jun 11 2016 *)
  • PARI
    Vec((3+2*x-9*x^2)/(1+x-9*x^2-x^3) + O(x^30)) \\ Colin Barker, Jun 11 2016
  • PARI
    polsym(x^3 + x^2 - 9*x - 1, 30) \\ Charles R Greathouse IV, Jul 20 2016

Formula

a(n) = (tan(Pi/7)/tan(4*Pi/7))^n + (tan(4*Pi/7)/tan(2*Pi/7))^n + (tan(2*Pi/7)/tan(Pi/7))^n.
a(n) = -a(n-1) + 9*a(n-2) + a(n-3) for n>2.
G.f.: (3+2*x-9*x^2) / (1+x-9*x^2-x^3). - Colin Barker, Jun 11 2016

A274220 a(n) = (-cos(Pi/7)/cos(2*Pi/7))^n + (-cos(2*Pi/7)/cos(3*Pi/7))^n + (cos(3*Pi/7)/cos(Pi/7))^n.

Original entry on oeis.org

3, -4, 10, -25, 66, -179, 493, -1369, 3818, -10672, 29865, -83626, 234237, -656205, 1838483, -5151080, 14432666, -40438941, 113306686, -317477255, 889550021, -2492461633, 6983719214, -19567941936, 54828148469, -153625048854, 430447808073, -1206087937261, 3379383275971, -9468821484028
Offset: 0

Views

Author

Kai Wang, Jun 14 2016

Keywords

Comments

a(n) is an integer.
This is other half of A215076.
a(n) is the sum of n-th powers of the roots of x^3 + 4*x^2 + 3*x - 1. - Greg Dresden, Mar 11 2020

Examples

			a(0) = 3, a(1) = -4, a(2) = 10, a(3) = -25.

References

  • R. Witula, E. Hetmaniok, D. Slota, Sums of the powers of any order roots taken from the roots of a given polynomial, Proceedings of the Fifteenth International Conference on Fibonacci Numbers and Their Applications, Eger, Hungary, 2012.

Links

Crossrefs

Cf. A215076, A033304, A094648, A274032.

Programs

  • Mathematica
    CoefficientList[Series[(3 + 8 x + 3 x^2)/(1 + 4 x + 3 x^2 - x^3), {x, 0, 29}], x] (* Michael De Vlieger, Jun 14 2016 *)
  • PARI
    Vec((3+8*x+3*x^2)/(1+4*x+3*x^2-x^3) + O(x^30)) \\ Colin Barker, Jun 14 2016
  • PARI
    polsym(x^3 + 4*x^2 + 3*x - 1,33) \\ Joerg Arndt, Mar 12 2020

Formula

a(n) = -4*a(n-1)-3*a(n-2)+a(n-3).
G.f.: (3+8*x+3*x^2) / (1+4*x+3*x^2-x^3). - Colin Barker, Jun 14 2016

Extensions

Many terms corrected by Colin Barker, Jun 14 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

Views

Author

Kai Wang, Jun 29 2016

Keywords

Comments

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)).

Links

Crossrefs

Cf. A248417, A274032, A274075.

Programs

  • Mathematica
    LinearRecurrence[{31,25,1},{3,31,1011},20] (* Harvey P. Dale, Feb 02 2022 *)
  • PARI
    Vec((3-62*x-25*x^2)/(1-31*x-25*x^2-x^3) + O(x^20)) \\ Colin Barker, Jun 30 2016
  • PARI
    polsym(x^3 -31* x^2 - 25*x - 1, 30) \\ Charles R Greathouse IV, Jul 20 2016

Formula

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 + 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

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

Views

Author

Kai Wang, May 24 2017

Keywords

Links

Crossrefs

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

Programs

  • Mathematica
    LinearRecurrence[{56,-784,3136},{3,56,1568},30] (* Harvey P. Dale, Aug 08 2017 *)
  • PARI
    Vec((3 - 28*x)*(1 - 28*x) / (1 - 56*x + 784*x^2 - 3136*x^3) + O(x^30)) \\ Colin Barker, May 25 2017
  • PARI
    polsym(x^3 - 56*x^2 + 784* x - 3136, 20) \\ Joerg Arndt, May 26 2017

Formula

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(1*Pi/7))^2)^n + (7*(cot(2*Pi/7))^2)^n + (7*(cot(4*Pi/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

Views

Author

Kai Wang, May 24 2017

Keywords

Comments

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.

Links

Crossrefs

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

Programs

  • Mathematica
    LinearRecurrence[{35,-147,49},{3,35,931},30] (* Harvey P. Dale, Mar 15 2018 *)
  • PARI
    Vec((3 - 7*x)*(1 - 21*x) / (1 - 35*x + 147*x^2 - 49*x^3) + O(x^30)) \\ Colin Barker, May 26 2017
  • PARI
    polsym(x^3 - 35*x^2 + 147*x - 49, 20) \\ Joerg Arndt, May 26 2017

Formula

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

A322455 Sum of n-th powers of the roots of x^3 - 20*x^2 - 9*x - 1.

Original entry on oeis.org

3, 20, 418, 8543, 174642, 3570145, 72983221, 1491970367, 30499826474, 623497246004, 12745935328713, 260560681614770, 5326550547499821, 108888803019858063, 2225975576006274419, 45504837297851710768, 930239414944110543194, 19016557810138882535211
Offset: 0

Views

Author

Kai Wang, Dec 09 2018

Keywords

Comments

Let A = sin(2*Pi/7), B = sin(4*Pi/7), C = sin(8*Pi/7).
In general, for integer h, k let
X = (B^h*C^k)/A^(h+k),
Y = (C^h*A^k)/B^(h+k),
Z = (A^h*B^k)/C^(h+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) and its other half is A320918.

Links

Crossrefs

Similar sequences with (h,k) values: A033304 (0,1), A215076 (1,1), A274032 (1,2).

Programs

  • Mathematica
    CoefficientList[Series[(3 - 40*x - 9*x^2) / (1 - 20*x - 9*x^2 - x^3) , {x, 0, 50}], x] (* Amiram Eldar, Dec 09 2018 *)
  • PARI
    Vec((3 - 40*x - 9*x^2) / (1 - 20*x - 9*x^2 - x^3) + O(x^20)) \\ Colin Barker, Dec 09 2018
  • PARI
    polsym(x^3 - 20*x^2 - 9*x - 1, 25) \\ Joerg Arndt, Dec 17 2018

Formula

a(n) = (B*C^3/A^4)^n + (C*A^3/B^4)^n + (A*B^3/C^4)^n.
a(n) = 20*a(n-1) + 9*a(n-2) + a(n-3) for n > 2.
G.f.: (3 - 40*x - 9*x^2) / (1 - 20*x - 9*x^2 - x^3). - Colin Barker, Dec 09 2018
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