A275830 -id:A275830 - cp's OEIS frontend

A215493 a(n) = 7a(n-1) - 14a(n-2) + 7*a(n-3) with a(0)=0, a(1)=1, a(2)=4.

0, 1, 4, 14, 49, 175, 637, 2352, 8771, 32928, 124166, 469567, 1779141, 6749211, 25623472, 97329337, 369821228, 1405502182, 5342323441, 20307982135, 77201862045, 293497548512, 1115812645899, 4242135876440, 16128056932078, 61317184775679, 233122447515741

Offset: 0

Roman Witula, Aug 13 2012

The Berndt-type sequence number 4 for the argument 2Pi/7 - see also A215007, A215008, A215143 and A215494.

We have a(n)=A079309(n) for n=1..6, and A079309(7)-a(7)=1.

G. C. Greubel, Table of n, a(n) for n = 0..1000
B. C. Berndt, A. Zaharescu, Finite trigonometric sums and class numbers, Math. Ann. 330 (2004), 551-575.
B. C. Berndt, L.-C. Zhang, Ramanujan's identities for eta-functions, Math. Ann. 292 (1992), 561-573.
Z.-G. Liu, Some Eisenstein series identities related to modular equations of the seventh order, Pacific J. Math. 209 (2003), 103-130.
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 (7,-14,7).

I:=[0,1,4]; [n le 3 select I[n] else 7*Self(n-1) - 14*Self(n-2) +7*Self(n-3): n in [1..30]]; // G. C. Greubel, Apr 23 2018

LinearRecurrence[{7,-14,7}, {0,1,4}, 50]

x='x+O('x^30); concat([0], Vec(x*(1-3*x)/(1-7*x+14*x^2-7*x^3))) \\ G. C. Greubel, Apr 23 2018

a(n)*sqrt(7) = s(1)^(2n-1) + s(2)^(2n-1) + s(4)^(2n-1), where s(j) := 2*Sin(2*Pi*j/7) (for the sums of the respective even powers see A215494, see also A094429, A115146). For the proof of these formula see Witula-Slota's paper.
G.f.: x*(1-3*x)/(1-7*x+14*x^2-7*x^3).
a(n) = A275830(2*n-1)/(7^n). - Kai Wang, May 25 2017

A275831 a(n) = (sqrt(7)csc(Pi/7)/2)^n + (-sqrt(7)csc(2Pi/7)/2)^n + (-sqrt(7)csc(4*Pi/7)/2)^n.

Original entry on oeis.org

3, 0, 14, 21, 98, 245, 833, 2401, 7546, 22638, 69629, 211288, 645869, 1966419, 6000099, 18286016, 55765626, 170002805, 518361494, 1580379017, 4818550093, 14691183577, 44792503770, 136568135690, 416385811429, 1269524476220, 3870677629833, 11801372013543, 35981414742371, 109704347503632, 334479507291398

Offset: 0

Kai Wang, Aug 11 2016

(sqrt(7)*csc(Pi/7)/2), (-sqrt(7)*csc(2*Pi/7)/2) and (-sqrt(7)*csc(4*Pi/7)/2) are the roots of the polynomial x^3 - 7*x - 7. - Corrected by Colin Barker, Aug 12 2016

Colin Barker, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (0,7,7).

Cf. A274975, A275195, A275830.

RecurrenceTable[{a[0] == 3, a[1] == 0, a[2] == 14, a[n] == 7 a[n - 2] + 7 a[n - 3]}, a, {n, 0, 30}] (* Bruno Berselli, Aug 11 2016 *)
LinearRecurrence[{0,7,7},{3,0,14},40] (* Harvey P. Dale, Jan 01 2022 *)

Vec((3-7*x^2)/(1-7*x^2-7*x^3) + O(x^30)) \\ Colin Barker, Aug 12 2016

G.f.: (3 - 7*x^2)/(1 - 7*x^2 - 7*x^3). - Bruno Berselli, Aug 11 2016

a(n) = 7*a(n-2) + 7*a(n-3) with n>2, a(0)=3, a(1)=0, a(2)=14.

Name and comment corrected by Colin Barker, Aug 12 2016

A322458 Sum of n-th powers of the roots of x^3 - 49*x + 49.

Original entry on oeis.org

3, 0, 98, -147, 4802, -12005, 242501, -823543, 12470794, -52236156, 651422513, -3170640550, 34479274781, -187281090087, 1844845851219, -10866257878532, 99574220123994, -622844082757799, 5411583422123774, -35398496841207857, 295686947739197077, -1999693932903249919

Offset: 0

Kai Wang, Dec 09 2018

Let A = sin(2*Pi/7), B = sin(4*Pi/7), C = sin(8*Pi/7).
In general, for integer 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 (0,49,-49).

Similar sequences with (h,k) values: A275830 (0,0).

LinearRecurrence[{0, 49, -49}, {3, 0, 98}, 50] (* Amiram Eldar, Dec 09 2018 *)

Vec((3 - 49*x^2) / (1 - 49*x^2 + 49*x^3) + O(x^25)) \\ Colin Barker, Dec 09 2018

polsym(x^3 - 49*x + 49, 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 = sin(2*Pi/7), B = sin(4*Pi/7), C = sin(8*Pi/7).
a(n) = 49*a(n-2) - 49 a(n-3) for n>2.
G.f.: (3 - 49*x^2) / (1 - 49*x^2 + 49*x^3). - Colin Barker, Dec 09 2018

cp's OEIS Frontend

A215493 a(n) = 7a(n-1) - 14a(n-2) + 7*a(n-3) with a(0)=0, a(1)=1, a(2)=4.

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A275831 a(n) = (sqrt(7)csc(Pi/7)/2)^n + (-sqrt(7)csc(2Pi/7)/2)^n + (-sqrt(7)csc(4*Pi/7)/2)^n.

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A322458 Sum of n-th powers of the roots of x^3 - 49*x + 49.

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cp's OEIS Frontend

A215493 a(n) = 7*a(n-1) - 14*a(n-2) + 7*a(n-3) with a(0)=0, a(1)=1, a(2)=4.

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Magma

Mathematica

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Formula

A275831 a(n) = (sqrt(7)*csc(Pi/7)/2)^n + (-sqrt(7)*csc(2*Pi/7)/2)^n + (-sqrt(7)*csc(4*Pi/7)/2)^n.

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A322458 Sum of n-th powers of the roots of x^3 - 49*x + 49.

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Author

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Programs

Mathematica

PARI

PARI

Formula

A215493 a(n) = 7a(n-1) - 14a(n-2) + 7*a(n-3) with a(0)=0, a(1)=1, a(2)=4.

A275831 a(n) = (sqrt(7)csc(Pi/7)/2)^n + (-sqrt(7)csc(2Pi/7)/2)^n + (-sqrt(7)csc(4*Pi/7)/2)^n.