A033304 -id:A033304 - cp's OEIS frontend

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.

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

A062882 a(n) = (1 - 2cos(Pi/9))^n + (1 + 2cos(Pi2/9))^n + (1 + 2cos(Pi*4/9))^n.

Original entry on oeis.org

3, 9, 18, 45, 108, 270, 675, 1701, 4293, 10854, 27459, 69498, 175932, 445419, 1127763, 2855493, 7230222, 18307377, 46355652, 117376290, 297206739, 752553261, 1905530913, 4824972522, 12217257783, 30935180610, 78330624264

Offset: 1

Vladeta Jovovic, Jun 27 2001

From L. Edson Jeffery, Apr 05 2011: (Start)
Let U be the matrix (see [Jeffery])
U = U_(9,2) =
  (0 0 1 0)
  (0 1 0 1)
  (1 0 1 1)
  (0 1 1 1).
Then a(n) = Trace(U^n).
(End)
We note that all numbers of the form a(n)*3^(-floor((n+4)/3)) are integers. - Roman Witula, Sep 29 2012

			We have a(2)=3*a(1), a(4)/a(3) = a(6)/a(5) = a(7)/a(6) = 5/2, a(6)=6*a(4), a(7)=15*a(4), and (1 + c(1))^8 + (1 + c(2))^8 + (1 + c(4))^8 = 7*3^5. - _Roman Witula_, Sep 29 2012

Harry J. Smith, Table of n, a(n) for n = 1..200
L. Edson Jeffery, Danzer matrices
Index entries for linear recurrences with constant coefficients, signature (3,0,-3).

Cf. A033304, A062883.

Digits := 1000:q := seq(floor(evalf((1-2*cos(1/9*Pi))^n+(1+2*cos(2/9*Pi))^n+(1+2*cos(4/9*Pi))^n)),n=1..50);

LinearRecurrence[{3,0,-3},{3,9,18},25] (* Georg Fischer Feb 02 2019 *)

{ default(realprecision, 200); for (n=1, 200, a=(1 - 2*cos(1/9*Pi))^n + (1 + 2*cos(2/9*Pi))^n + (1 + 2*cos(4/9*Pi))^n; write("b062882.txt", n, " ", round(a)) ) } \\ Harry J. Smith, Aug 12 2009

Vec((3-9*x^2)/(1-3*x+3*x^3)+O(x^66)) /* Joerg Arndt, Apr 08 2011 */

G.f.: x*(3 - 9*x^2)/(1 - 3*x + 3*x^3). The terms in parentheses in the definition are the roots of x^3-3*x^2+3. - Ralf Stephan, Apr 10 2004

a(n) = 3*(a(n-1) - a(n-3)) for n >= 4 - Roman Witula, Sep 29 2012

More terms from Sascha Kurz, Mar 24 2002
Adapted formula, denominator of g.f. and modified g.f. (and offset) to accommodate added initial term a(0)=4. - L. Edson Jeffery, Apr 05 2011
a(0) = 4 removed, g.f. and programs adapted by Georg Fischer, Feb 02 2019

A321173 a(n) = -2*a(n-1) + a(n-2) + a(n-3), a(0) = -1, a(1) = 3, a(2) = -9.

Original entry on oeis.org

-1, 3, -9, 20, -46, 103, -232, 521, -1171, 2631, -5912, 13284, -29849, 67070, -150705, 338631, -760897, 1709720, -3841706, 8632235, -19396456, 43583441, -97931103, 220049191, -494446044, 1111010176, -2496417205, 5609398542, -12604204113, 28321389563, -63637584697

Offset: 0

Kai Wang, Jan 10 2019

Let {X,Y,Z} be the roots of the cubic equation t^3 + at^2 + bt + c = 0 where {a, b, c} are integers.
Let {u, v, w} be three numbers such that {u + v + w, u*X + v*Y + w*Z, u*X^2 + v*Y^2 + w*Z^2} are integers.
Then {p(n) = u*X^n + v*Y^n + w*Z^n | n = 0, 1, 2, ...} is an integer sequence with the recurrence relation: p(n) = -a*p(n-1) - b*p(n-2) - c*p(n-3).
Let k = Pi/7.
This sequence has (a, b, c) = (2, -1, -1), (u, v, w) = (2*cos(4k), 2*cos(8k), 2*cos(2k)).
A033304, A274975: (a, b, c) = (2, -1, -1), (u, v, w) = (2*cos(2k), 2*cos(4k), 2*cos(8k)).
A321174 : (a, b, c) = (2, -1, -1), (u, v, w) = (2*cos(8k), 2*cos(2k), 2*cos(4k)).
X = sin(2k)/sin(4k), Y = sin(4k)/sin(8k), Z = sin(8k)/sin(2k).

Colin Barker, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (-2,1,1).

Cf. A321174, A033304, A274975.

LinearRecurrence[{-2,1,1},{-1,3,-9},50] (* Stefano Spezia, Jan 11 2019 *)

Vec(-(1 - x + 2*x^2) / (1 + 2*x - x^2 - x^3) + O(x^30)) \\ Colin Barker, Jan 11 2019

G.f.: -(1 - x + 2*x^2) / (1 + 2*x - x^2 - x^3). - Colin Barker, Jan 11 2019

A321174 a(n) = -2*a(n-1) + a(n-2) + a(n-3), a(0) = -1, a(1) = -4, a(2) = 5.

Original entry on oeis.org

-1, -4, 5, -15, 31, -72, 160, -361, 810, -1821, 4091, -9193, 20656, -46414, 104291, -234340, 526557, -1183163, 2658543, -5973692, 13422764, -30160677, 67770426, -152278765, 342167279, -768842897, 1727574308, -3881824234, 8722379879, -19599009684, 44038575013

Offset: 0

Kai Wang, Jan 10 2019

Let {X,Y,Z} be the roots of the cubic equation t^3 + at^2 + bt + c = 0 where {a, b, c} are integers.
Let {u, v, w} be three numbers such that {u + v + w, u*X + v*Y + w*Z, u*X^2 + v*Y^2 + w*Z^2} are integers.
Then {p(n) = u*X^n + v*Y^n + w*Z^n | n = 0, 1, 2, ...} is an integer sequence with the recurrence relation: p(n) = -a*p(n-1) - b*p(n-2) - c*p(n-3).
Let k = Pi/7.
This sequence has (a, b, c) = (2, -1, -1), (u, v, w) = (2*cos(8k), 2*cos(2k), 2*cos(4k)).
A033304, A274975: (a, b, c) = (2, -1, -1), (u, v, w) = (2*cos(2k), 2*cos(4k), 2*cos(8k)).
A321173 : (a, b, c) = (2, -1, -1), (u, v, w) = (2*cos(4k), 2*cos(8k), 2*cos(2k)).
X = sin(2k)/sin(4k), Y = sin(4k)/sin(8k), Z = sin(8k)/sin(2k).

Colin Barker, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (-2,1,1).

Cf. A321173, A033304, A274975.

LinearRecurrence[{-2,1,1},{-1,-4,5},50] (* Stefano Spezia, Jan 11 2019 *)

Vec(-(1 + 6*x + 2*x^2) / (1 + 2*x - x^2 - x^3) + O(x^30)) \\ Colin Barker, Jan 11 2019

G.f.: -(1 + 6*x + 2*x^2) / (1 + 2*x - x^2 - x^3). - Colin Barker, Jan 11 2019

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

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

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 = (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.

Colin Barker, Table of n, a(n) for n = 0..750
Index entries for linear recurrences with constant coefficients, signature (20,9,1).

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

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 *)

Vec((3 - 40*x - 9*x^2) / (1 - 20*x - 9*x^2 - x^3) + O(x^20)) \\ Colin Barker, Dec 09 2018

polsym(x^3 - 20*x^2 - 9*x - 1, 25) \\ Joerg Arndt, Dec 17 2018

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

A218664 Coefficients of cubic polynomials p(x+n), where p(x) = x^3 + x^2 - 2*x - 1.

Original entry on oeis.org

1, 1, -2, -1, 1, 4, 3, -1, 1, 7, 14, 7, 1, 10, 31, 29, 1, 13, 54, 71, 1, 16, 83, 139, 1, 19, 118, 239, 1, 22, 159, 377, 1, 25, 206, 559, 1, 28, 259, 791, 1, 31, 318, 1079, 1, 34, 383, 1429, 1, 37, 454, 1847, 1, 40, 531, 2339, 1, 43, 614, 2911, 1, 46, 703, 3569, 1, 49, 798, 4319

Offset: 0

Roman Witula, Nov 04 2012

sign

We have p(x) = (x - c(1))*(x - c(2))*(x - c(4)), where c(j) := 2*cos(2*Pi*j/7). We note that c(4) = c(3) = -c(1/2), c(1) = s(3) and c(2) = -s(1), where s(j) := 2*sin(Pi*j/14). Moreover we obtain -p(-x) = x^3 - x^2 - 2*x + 1 = (x + c(1))*(x + c(2))*(x + c(4)), q(x) := -x^3*p(1/x) = x^3 + 2*x^2 + x - 1 = (x - c(1)^(-1))*(x - c(2)^(-1))*(x - c(4)^(-1)), and -q(-x) = x^3 - 2*x^2 + x + 1 = (x + c(1)^(-1))*(x + c(2)^(-1))*(x + c(4)^(-1)).
We also have p(x+2) =  x^3 + 7*x^2 + 14*x + 7 = (x + s(2)^2)*(x + s(4)^2)*(x + s(6)^2). The polynomial -p(-x-2) = x^3 - 7*x^2 + 14*x - 7 = (x - s(2)^2)*(x - s(4)^2)*(x - s(6)^2) is known as Johannes Kepler's cubic polynomial (see Witula's book).
Let us set r(x) := p(x+1). It can be verified that -x^3*r(1/x) = x^3 - 3*x^2 - 4*x - 1 = (x - c(1)/c(4))*(x - c(4)/c(2))*(x - c(2)/c(1)); for example, we have c(1)^3 + c(1)^2 - 2*c(1) - 1 = 0 which implies that c(1)^2 + 2*c(1) = 1/(c(1) - 1), and then c(1)^2 + 2*c(1) = c(4)/c(2) since c(4)/c(2) = (c(1)^4 - 4*c(1)^2 + 2)/(c(1)^2 - 2).
The polynomials p(x+n) and the ones obtained as above (i.e., after simple algebraic transformations) are the characteristic polynomials of many sequences in the OEIS; see crossrefs.

R. Witula, Complex Numbers, Polynomials and Partial Fraction Decomposition, Part 3, Wydawnictwo Politechniki Slaskiej, Gliwice 2010 (Silesian Technical University publishers).

Cf. A094648, A096975, A033304, A214683, A006053, A215076, A215100, A215007, A215008, A215143, A215493, A215494, A215510.

We have a(4*k) = 1, a(4*k + 1) = 3*k + 1, a(4*k + 2) = 3*k^2 + 2*k - 2, a(4*k + 3) = k^3 + k^2 - 2*k - 1. Further, the following relations hold true: b(k+1) = b(k) + 3, c(k+1) = 2*b(k) -2*c(k) + 3, d(k+1) = b(k) - 2*c(k) - d(k) + 1, where p(x + k) = x^3 + b(k)*x^2 + c(k)*x + d(k).

Empirical g.f.: -(x^15 - x^14 - 2*x^13 + x^12 - 5*x^11 + 10*x^10 + 3*x^9 - 3*x^8 - 3*x^7 - 11*x^6 + 3*x^4 + x^3 + 2*x^2 - x - 1) / ((x-1)^4*(x+1)^4*(x^2+1)^4). - Colin Barker, May 17 2013

cp's OEIS Frontend

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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A062882 a(n) = (1 - 2*cos(Pi/9))^n + (1 + 2*cos(Pi*2/9))^n + (1 + 2*cos(Pi*4/9))^n.

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A321173 a(n) = -2*a(n-1) + a(n-2) + a(n-3), a(0) = -1, a(1) = 3, a(2) = -9.

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A321174 a(n) = -2*a(n-1) + a(n-2) + a(n-3), a(0) = -1, a(1) = -4, a(2) = 5.

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

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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.

A062882 a(n) = (1 - 2cos(Pi/9))^n + (1 + 2cos(Pi2/9))^n + (1 + 2cos(Pi*4/9))^n.

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.

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