A215575 -id:A215575 - cp's OEIS frontend

A215945 a(n) = - 3^nA(2n+1), where A(n) = 3*A(n-1) + A(n-2) - A(n-3)/3, with A(0)=A(1)=3, A(2)=11.

-3, -105, -3387, -108945, -3504051, -112702329, -3624894315, -116589061665, -3749904995427, -120609834867081, -3879226882922139, -124769271310005681, -4013008656895890963, -129072153032843014809, -4151404124161560449739

Offset: 0

Roman Witula, Aug 28 2012

The Berndt-type sequence number 11 for the argument 2Pi/9 connecting with the following sequence T(n) := t(1)^n + (-t(2))^n + t(4)^n, where t(j) := tan(2*Pi*j/9). More precisely we have sqrt(3)*a(n) = T(2*n+1) = (-sqrt(3))^n*W(n;2*i/sqrt(3)), where W(n;d) := (1 + 2*i*d*s(1))^n
+ (1 - 2*i*d*s(2))^n + (1 + 2*i*d*s(4))^n, n=0,1,..., d in C. For example we have W(0;d)=W(1;d)=3, W(2;d)=3-6*d^2. The characteristic polynomial of W(n;d) has the form ((x-1)-2*i*d*s(1))*((x-1)-2*i*d*s(2))*((x-1)-2*i*d*s(4)) = (x-1)^3 + 3*d^2*(x-1) - sqrt(3)*i*d^3 = x^3 - 3*x^2 + 3*(1+d^2)*x - 1 - 3*d^2 - sqrt(3)*i*d^3, which implies the recurrence form of W(n;d) = 3*W(n-1;d) - 3*(1+d^2)*W(n-2;d) + (1+3*d^2+sqrt(3)*i*d^3)*W(n-3;d). In consequence we obtain the title recurrence relation for
A(n) := W(n;2*i/sqrt(3)). The polynomials W(n;d) are equivalent of the big omega function with index n of argument d defined and discussed in Witula-Slota's paper (Section 6) and in comments to A215794.
We note that all numbers a(n)/3 and 3^(-1+floor((n+1)/3))*A(n) = A216034(n) are integers.
The following decomposition hold true: (X - t(1)^n)*(X - (-t(2))^n)*(X - t(4)^n) = X^3 - (-sqrt(3))^n*A(n)*X^2 + A215829(n)*X - sqrt(3)^n. Moreover we have A215829(n) = (T(n)^2 - T(2*n))/2, which implies A215829(2*n+1) = (3*a(n)^2 - A215948(2*n+1))/2 and A215829(2*n) = (A215948(n)^2 - A215948(2*n))/2.

			We have 35*(t(1) - t(2) + t(3)) =  t(1)^3 - t(2)^3 + t(4)^3, t(1)^7 - t(2)^7 + t(4)^7 = -5*81*269*sqrt(3) and t(1)^9 - t(2)^9 + t(4)^9 = -9*389339*sqrt(3).

D. Chmiela and R. Witula, Two parametric quasi-Fibonacci numbers of the ninth order, (submitted, 2012).
R. Witula, Ramanujan type formulas for arguments 2Pi/7 and 2Pi/9, Demonstratio Math. (in press, 2012).

Vincenzo Librandi, Table of n, a(n) for n = 0..600
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 (33,-27,3).

Cf. A215948, A216034, A215829, A215794, A215575.

m:=17; R:=PowerSeriesRing(Integers(), m); Coefficients(R!(-3*(1+x)^2/(1-33*x+27*x^2-3*x^3))); // Bruno Berselli, Aug 29 2012

I:=[-3, -105, -3387]; [n le 3 select I[n] else 33*Self(n-1)-27*Self(n-2)+3*Self(n-3): n in [1..20]]; // Vincenzo Librandi, Mar 19 2013

LinearRecurrence[{33, -27, 3}, {-3, -105, -3387}, 17] (* Bruno Berselli, Aug 29 2012 *)
CoefficientList[Series[-3 (1 + x)^2/(1 - 33 x + 27 x^2 - 3 x^3), {x, 0, 20}], x] (* Vincenzo Librandi, Mar 19 2013 *)

sqrt(3)*a(n) = t(1)^(2*n+1) - t(2)^(2*n+1) + t(4)^(2*n+1) = (-sqrt(3) + 4*s(1))^(2*n+1) + (-sqrt(3) - 4*s(2))^(2*n+1) + (-sqrt(3) + 4*s(4))^(2*n+1), where t(j) := tan(2*Pi*j/9) and s(j) := sin(2*Pi*j/9).
a(n) = 33*a(n-1) - 27*a(n-2) + 3*a(n-3).
G.f.: -3*(1+x)^2/(1-33*x+27*x^2-3*x^3). - Bruno Berselli, Aug 29 2012

A215948 a(n) = 3^nA(2n), where A(n) = 3*A(n-1) + A(n-2) - A(n-3)/3 with A(0)=A(1)=3, A(2)=11.

Original entry on oeis.org

3, 33, 1035, 33273, 1070163, 34420113, 1107069147, 35607149289, 1145248319907, 36835122733569, 1184744167018155, 38105444942752473, 1225602095969542131, 39419576386041628017, 1267869080483024344443, 40779027899804588036553, 1311593714249667872790339

Offset: 0

Roman Witula, Aug 28 2012

The Berndt-type sequence number 12 for the argument 2*Pi/9 defined by the first trigonometric relations from the section "Formula" below (it is the complement of the sequence A215945). For more information see comments to A215945. We note that all a(n)/3 and 3^(-1 + floor((n+3)/3))*A(n) = A216034(n) are integers.

			We have t(1)^4 + t(2)^4 + t(4)^4 = 1035 = (345/11)*(t(1)^2 + t(2)^2 + t(4)^2) and (1 - 4*s(1)/sqrt(3))^4 + (1 + 4*s(2)/sqrt(3))^4 + (1 - 4*s(4)/sqrt(3))^4 = 115. Moreover we get a(2)/a(1) = 31,(36), a(3)/a(1) = 1008,(27), a(4)/a(1) = 32429,(18).

D. Chmiela and R. Witula, Two parametric quasi-Fibonacci numbers of the ninth order, (submitted, 2012).
R. Witula, Ramanujan type formulas for arguments 2Pi/7 and 2Pi/9, Demonstratio Math. (in press, 2012).

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 (33,-27,3).

Cf. A215945, A216034, A215829, A215794, A215575.

LinearRecurrence[{33,-27,3}, {3,33,1035}, 50]

a(n) = t(1)^(2*n) + t(2)^(2*n) + t(4)^(2*n) = (-sqrt(3) + 4*s(1))^(2*n) + (sqrt(3) + 4*s(2))^(2*n) + (-sqrt(3) + 4*s(4))^(2*n), where t(j) := tan(2*Pi*j/9) and s(j) := sin(2*Pi*j/9). For the respective sums of odd powers - see A215945.
a(n) = 33*a(n-1) - 27*a(n-2) + 3*a(n-3).
G.f.: 3*(1-22*x+9*x^2)/(1-33*x+27*x^2-3*x^3).
a(n) = cot(Pi/18)^(2*n) + cot(5*Pi/18)^(2*n) + cot(7*Pi/18)^(2*n). - Greg Dresden, Oct 01 2020

A216034 a(n) = 3^(-1+floor((n+1)/3))A(n), where A(n) = 3A(n-1) + A(n-2) - A(n-3)/3 with A(0)=A(1)=3, A(2)=11.

Original entry on oeis.org

1, 1, 11, 35, 115, 1129, 3697, 12105, 118907, 389339, 1274819, 12522481, 41002561, 134255345, 1318783307, 4318113395, 14138868147, 138885370201, 454754601649, 1489010307001, 14626471197755, 47891689912619, 156812530628611, 1540361374197601

Offset: 0

Roman Witula, Aug 30 2012

The Berndt-type sequence number 11a for the argument 2Pi/9 - see A215945 for more details.

Index entries for linear recurrences with constant coefficients, signature (0,0,105,0,0,33,0,0,-1).

Cf. A215945, A215948, A215829, A215794, A215575.

i:=24; I:=[3,3,11]; A:=[m le 3 select I[m] else 3*Self(m-1)+Self(m-2)-Self(m-3)/3: m in [1..i]]; [3^(-1+Floor(n/3))*A[n]: n in [1..i]]; // Bruno Berselli, Oct 02 2012

G.f.: (1+x+11*x^2-70*x^3+10*x^4-26*x^5-11*x^6-3*x^7-x^8)/(1-105*x^3-33*x^6+x^9). [Bruno Berselli, Oct 02 2012]

A217444 a(n) = A(n)7^(-floor(n+1)/3), where A(n) = 7A(n-1) - 14A(n-2) + 7A(n-3) with A(0)=0, A(1)=1, A(2)=7.

Original entry on oeis.org

0, 1, 1, 5, 22, 13, 52, 204, 113, 435, 1667, 910, 3471, 13224, 7192, 27367, 104105, 56563, 215098, 817909, 444276, 1689212, 6422529, 3488381, 13262821, 50424942, 27387681, 104126704, 395884336, 215018609, 817488295, 3108041875, 1688083894, 6417991803, 24400809980

Offset: 0

Roman Witula, Oct 03 2012

The Berndt-type sequence number 18a for the argument 2Pi/7, which is closely connected with the sequence A217274. Definitions other Berndt-type sequences for the argument 2Pi/7 like A215575, A215877, A033304 in sequences from Crossrefs are given.

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (0,0,10,0,0,-17,0,0,1).

Cf. A215007, A215008, A215143, A215493, A215494, A215510, A215512, A215575, A215694, A215695, A108716, A215794, A215828, A215817, A215877, A094429, A094430, A217274, A094648, A033304.

i:=35; I:=[0, 1, 7]; A:=[m le 3 select I[m] else 7*Self(m-1)-14*Self(m-2)+7*Self(m-3): m in [1..i]]; [7^(-Floor(n/3))*A[n]: n in [1..i]]; // Bruno Berselli, Oct 03 2012

CoefficientList[Series[x*(1+x+5*x^2+12*x^3+3*x^4+2*x^5+x^6)/(1 - 10*x^3 + 17*x^6 - x^9), {x, 0, 40}], x] (* Vincenzo Librandi, Dec 15 2012 *)
LinearRecurrence[{0,0,10,0,0,-17,0,0,1}, {0, 1, 1, 5, 22, 13, 52, 204, 113}, 50] (* G. C. Greubel, Apr 23 2018 *)

x='x+O('x^30); concat([0], Vec(x*(1+x+5*x^2+12*x^3+3*x^4 +2*x^5 +x^6)/(1- 10*x^3+17*x^6-x^9))) \\ G. C. Greubel, Apr 23 2018

G.f.: x*(1+x+5*x^2+12*x^3+3*x^4+2*x^5+x^6)/(1-10*x^3+17*x^6-x^9). - Bruno Berselli, Oct 03 2012

a(n) = 10*a(n-3) - 17*a(n-6) + a(n-9). - G. C. Greubel, Apr 23 2018

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

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 = 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).

Colin Barker, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (-7,-14,-7).

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

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 *)

Vec((3 + 14*x + 14*x^2) / (1 + 7*x + 14*x^2 + 7*x^3) + O(x^40)) \\ Colin Barker, Dec 09 2018

polsym(x^3 + 7*x^2 + 14*x + 7, 25) \\ Joerg Arndt, Dec 17 2018

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

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

cp's OEIS Frontend

A215945 a(n) = - 3^n*A(2*n+1), where A(n) = 3*A(n-1) + A(n-2) - A(n-3)/3, with A(0)=A(1)=3, A(2)=11.

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A215948 a(n) = 3^n*A(2*n), where A(n) = 3*A(n-1) + A(n-2) - A(n-3)/3 with A(0)=A(1)=3, A(2)=11.

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A216034 a(n) = 3^(-1+floor((n+1)/3))*A(n), where A(n) = 3*A(n-1) + A(n-2) - A(n-3)/3 with A(0)=A(1)=3, A(2)=11.

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A217444 a(n) = A(n)*7^(-floor(n+1)/3), where A(n) = 7*A(n-1) - 14*A(n-2) + 7*A(n-3) with A(0)=0, A(1)=1, A(2)=7.

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

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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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A215945 a(n) = - 3^nA(2n+1), where A(n) = 3*A(n-1) + A(n-2) - A(n-3)/3, with A(0)=A(1)=3, A(2)=11.

A215948 a(n) = 3^nA(2n), where A(n) = 3*A(n-1) + A(n-2) - A(n-3)/3 with A(0)=A(1)=3, A(2)=11.

A216034 a(n) = 3^(-1+floor((n+1)/3))A(n), where A(n) = 3A(n-1) + A(n-2) - A(n-3)/3 with A(0)=A(1)=3, A(2)=11.

A217444 a(n) = A(n)7^(-floor(n+1)/3), where A(n) = 7A(n-1) - 14A(n-2) + 7A(n-3) with A(0)=0, A(1)=1, A(2)=7.

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

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.