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 1-2 of 2 results.

A275830 a(n) = (2*sqrt(7)*sin(Pi/7))^n + (-2*sqrt(7)*sin(2*Pi/7))^n + (-2*sqrt(7)*sin(4*Pi/7))^n.

Original entry on oeis.org

3, -7, 49, -196, 1029, -4802, 24010, -117649, 588245, -2941225, 14823774, -74942413, 380476866, -1936973136, 9886633715, -50563069571, 259029803333, -1328763571296, 6823754590093, -35073821767334, 180407337377834, -928487386730281, 4780794440512601, -24625601552074341, 126883328914736618
Offset: 0

Views

Author

Kai Wang, Aug 11 2016

Keywords

Comments

2*sqrt(7)*sin(Pi/7), -2*sqrt(7)*sin(2*Pi/7) and -2*sqrt(7)*sin(4*Pi/7) are roots of polynomial x^3 + 7*x^2 - 49.

Links

Crossrefs

Cf. A108716, A215794, A275195, A274975, A275831.

Programs

  • Mathematica
    RecurrenceTable[{a[0] == 3, a[1] == -7, a[2] == 49, a[n] == -7 a[n - 1] + 49 a[n - 3]}, a, {n, 0, 30}] (* Bruno Berselli, Aug 11 2016 *)
  • PARI
    Vec((3 + 14*x)/(1 + 7*x - 49*x^3) + O(x^30)) \\ Colin Barker, Aug 30 2016

Formula

G.f.: (3 + 14*x)/(1 + 7*x - 49*x^3). - Bruno Berselli, Aug 11 2016
a(n) = -7*a(n-1) + 49*a(n-3) with n>2, a(0)=3, a(1)=-7, a(2)=49.
a(2*n-1) = 7^n*A215493(n). - Kai Wang, May 25 2017

A322459 Sum of n-th powers of the roots of x^3 + 7*x^2 + 14*x + 7.

Original entry on oeis.org

3, -7, 21, -70, 245, -882, 3234, -12005, 44933, -169099, 638666, -2417807, 9167018, -34790490, 132119827, -501941055, 1907443237, -7249766678, 27557748813, -104759610858, 398257159370, -1514069805269, 5756205681709, -21884262613787, 83201447389466, -316323894905207
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 = sqrt(7)*A^(h+k-1)/(2*B^h*C^k),
Y = sqrt(7)*B^(h+k-1)/(2*C^h*A^k),
Z = sqrt(7)*C^(h+k-1)/(2*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) = (1,1).

Links

Crossrefs

Similar sequences with (h,k) values: A275831 (0,0), A215575 (0,2).

Programs

  • Mathematica
    LinearRecurrence[{-7, -14, -7},{3, -7, 21}, 50] (* Amiram Eldar, Dec 09 2018 *)
CoefficientList[Series[(3+14*x+14*x^2)/(1+7*x+14*x^2+7*x^3), {x, 0, 25}], x] (* G. C. Greubel, Dec 16 2018 *)
  • PARI
    Vec((3 + 14*x + 14*x^2) / (1 + 7*x + 14*x^2 + 7*x^3) + O(x^40)) \\ Colin Barker, Dec 09 2018
  • PARI
    polsym(x^3 + 7*x^2 + 14*x + 7, 25) \\ Joerg Arndt, Dec 17 2018

Formula

a(n) = (sqrt(7))^n*( (A/(2*B*C))^n + (B/(2*C*A))^n + (C/(2*A*B))^n ).
a(n) = -7*a(n-1) - 14*a(n-2) - 7*a(n-3) for n>2.
G.f.: (3 + 14*x + 14*x^2) / (1 + 7*x + 14*x^2 + 7*x^3). - Colin Barker, Dec 09 2018
Showing 1-2 of 2 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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