A122600 -id:A122600 - cp's OEIS frontend

A215076 a(n) = 3a(n-1) + 4a(n-2) + a(n-3) with a(0)=3, a(1)=3, a(2)=17.

3, 3, 17, 66, 269, 1088, 4406, 17839, 72229, 292449, 1184102, 4794331, 19411850, 78596976, 318232659, 1288497731, 5217020805, 21123285998, 85526438945, 346289481632, 1402097486674, 5676976825495, 22985609904813, 93066834503093, 376819919954026, 1525712707779263

Offset: 0

Roman Witula, Aug 02 2012

We call the sequence a(n) the Ramanujan-type sequence number 3 for the argument 2Pi/7 (see A214683 and Witula's papers for details). Since a(n)=as(3n), bs(3n)=cs(3n)=0, where the sequence as(n) and its two conjugate sequences bs(n) and cs(n) are defined in the comments to the sequence A214683 we obtain the following formula a(n) = (c(1)/c(4))^n + (c(2)/c(1))^n + (c(4)/c(2))^n, where c(j) := Cos(2*Pi*j/7). It is interesting that if we set b(n):= (c(1)/c(2))^n + (c(2)/c(4))^n + (c(4)/c(1))^n, for n=0,1,..., and we extend the definition of discussed sequence a(n) to the negative indices by the same formula, i.e.: a(n)=a(n+3)-3*a(n+2)-4*a(n+1), n=-1,-2,..., then we get b(n)=a(-n) for every n=0,1,... (see also example below).

			We have (c(1)/c(2)) + (c(2)/c(4)) + (c(4)/c(1)) = (a(1)^2 - a(2))/2 = -4, and then (c(1)/c(2))^2 + (c(2)/c(4))^2 + (c(4)/c(1))^2 = 16 - 2*a(1) = 10.

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

G. C. Greubel, Table of n, a(n) for n = 0..1000
Roman Witula, Ramanujan Type Trigonometric Formulas: The General Form for the Argument 2*Pi/7, Journal of Integer Sequences, Vol. 12 (2009), Article 09.8.5.
Roman Witula, Full Description of Ramanujan Cubic Polynomials, Journal of Integer Sequences, Vol. 13 (2010), Article 10.5.7.
Roman Witula, Ramanujan Cubic Polynomials of the Second Kind, Journal of Integer Sequences, Vol. 13 (2010), Article 10.7.5.
Roman Witula, Ramanujan Type Trigonometric Formulae, Demonstratio Math. 45 (2012) 779-796.
Index entries for linear recurrences with constant coefficients, signature (3,4,1).

Cf. A214683.

Mathematica
```
LinearRecurrence[{3,4,1},{3,3,17},40]
```

Vec((-3+6*x+4*x^2)/(-1+3*x+4*x^2+x^3) + O(x^30)) \\ Michel Marcus, Apr 20 2016

polsym(1+4*x+3*x^2-x^3, 22) \\ Joerg Arndt, Jul 09 2020

@CachedFunction
def a(n): # a = A215076
    if (n<3): return (3,3,17)[n]
    else: return 3*a(n-1) + 4*a(n-2) + a(n-3)
[a(n) for n in range(40)] # G. C. Greubel, Nov 25 2022

G.f.: (3-6*x-4*x^2)/(1-3*x-4*x^2-x^3).
From Kai Wang, Jul 08 2020: (Start)
a(n) = Sum_{i+2j+3k=n} 3^i*4^j*n*(i+j+k-1)!/(i!*j!*k!).
a(n) = (-1)^n*(3*A122600(n) + 6*A122600(n-1) - 4*A122600(n-2)) for n > 1. (End)
a(n) = r^n + s^n + t^n where {r,s,t} are the roots of 1+4*x+3*x^2-x^3. - Joerg Arndt, Jul 09 2020
a(n) = 3*a(n-1) + 4*a(n-2) + a(n-3). - Wesley Ivan Hurt, Jul 09 2020

More terms from Michel Marcus, Apr 20 2016

A085810 Number of three-choice paths along a corridor of height 5, starting from the lower side.

Original entry on oeis.org

1, 2, 5, 13, 35, 96, 266, 741, 2070, 5791, 16213, 45409, 127206, 356384, 998509, 2797678, 7838801, 21963661, 61540563, 172432468, 483144522, 1353740121, 3793094450, 10628012915, 29779028189, 83438979561, 233790820762, 655067316176, 1835457822857, 5142838522138, 14409913303805

Offset: 1

Philippe Deléham, Jul 25 2003

From Svjetlan Feretic, Jun 01 2013: (Start)
A three-choice path is a path whose steps lie in the set {(1,1), (1,0), (1,-1)}.
The paths under consideration "live" in a corridor like 0<=y<=5. Thus, the ordinate of a vertex of a path can take six values (0,1,2,3,4,5), but the height of the corridor is five.
a(1)=1 is the number of paths with zero steps, a(2)=2 is the number of paths with one step, a(3)=5 is the number of paths with two steps, ...
Narrower corridors produce A000012, A000079, A000129, A001519, A057960. An infinitely wide corridor would produce A005773.
(End)
Diagonal sums of A114164. - Paul Barry, Nov 15 2005
C(n):= a(n)*(-1)^n appears in the following formula for the nonpositive powers of rho*sigma, where rho:=2*cos(Pi/7) and sigma:=sin(3*Pi/7)/sin(Pi/7) = rho^2-1 are the ratios of the smaller and larger diagonal length to the side length in a regular 7-gon (heptagon). See the Steinbach reference where the basis  <1,rho,sigma> is used in an extension of the rational field. (rho*sigma)^(-n) = C(n) + B(n)*rho + A(n)*sigma,n>=0, with B(n)= A181880(n-2)*(-1)^n, and A(n)= A116423(n+1)*(-1)^(n+1). For the nonnegative powers see A120757(n), |A122600(n-1)| and A181879(n), respectively. See also a comment under A052547.
a(n) is also the number of bi-wall directed polygons with n cells. (The definition of bi-wall directed polygons is given in the article on A122737.)

G. C. Greubel, Table of n, a(n) for n = 1..1000
Svjetlan Feretic, Generating functions for bi-wall directed polygons, in: Proc. of the Seventh Int. Conf. on Lattice Path Combinatorics and Applications (eds. S. Rinaldi and S. G. Mohanty), Siena, 2010, 147-151.
Peter Steinbach, Golden fields: a case for the heptagon, Math. Mag. 70 (1997), p. 22-31 (formula 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 (4,-3,-1).

Cf. A000012, A000079, A000129, A001519, A057960, A005773.

I:=[1,2,5]; [n le 3 select I[n] else 4*Self(n-1)-3*Self(n-2)-Self(n-3): n in [1..35]]; // Vincenzo Librandi, Sep 18 2015

LinearRecurrence[{4,-3,-1}, {1,2,5}, 50] (* Roman Witula, Aug 09 2012 *)
CoefficientList[Series[(1 - 2 x)/(1 - 4 x + 3 x^2 + x^3), {x, 0, 40}], x] (* Vincenzo Librandi, Sep 18 2015 *)

x='x+O('x^30); Vec((1-2*x)/(1-4*x+3*x^2+x^3)) \\ G. C. Greubel, Apr 19 2018

a(n) = 4*a(n-1) - 3*a(n-2) - a(n-3).
From Paul Barry, Nov 15 2005: (Start)
G.f.: (1-2*x)/(1-4*x+3*x^2+x^3).
a(n) = Sum_{k=0..floor(n/2)} (Sum_{j=0..n-k} C(n-k, j)*C(j+k, 2k));
a(n) = Sum_{k=0..floor(n/2)} (Sum_{j=0..n-k} C(n-k, k+j)*C(k, k-j)*2^(n-2k-j));
a(n) = Sum_{k=0..floor(n/2)} (Sum_{j=0..n-2*k} C(n-j, n-2*k-j)*C(k, j)(-1)^j*2^(n-2*k-j)). (End)
a(n-1) = -B(n;-1) = (1/7)*((c(4)-c(1))*(1-c(1))^n + (c(1)-c(2))*(1-c(2))^n + (c(2)-c(4))*(1-c(4))^n), where a(-1):=0, c(j):=2*cos(2*Pi*j/7). Moreover, B(n;d), n in N, d in C, denotes the respective quasi-Fibonacci number defined in comments to A121449 or in Witula-Slota-Warzynski's paper (see also A077998, A006054, A052975, A094789, A121442). - Roman Witula, Aug 09 2012

Name corrected and clarified, and offset 1 from Svjetlan Feretic, Jun 01 2013

A120757 Expansion of x^2(2+x)/(1-3x-4*x^2-x^3).

Original entry on oeis.org

0, 2, 7, 29, 117, 474, 1919, 7770, 31460, 127379, 515747, 2088217, 8455018, 34233669, 138609296, 561217582, 2272323599, 9200450421, 37251863241, 150829715006, 610697048403, 2472661868474, 10011603514040, 40536155064419

Offset: 1

Gary W. Adamson & Roger L. Bagula, Jul 01 2006

The (1,1)-entry of the matrix M^n, where M is the 3 X 3 matrix [0,1,1; 1,1,2; 1,2,2].
a(n)/a(n-1) tends to 4.0489173...an eigenvalue of M and a root to the characteristic polynomial x^3 - 3x^2 - 4x - 1.
C(n):=a(n), with a(0):=1 (hence the o.g.f. for C(n) is (1-3*x-2*x^2)/(1-3*x-4*x^2-x^3)), appears in the following formula for the nonnegative powers of rho*sigma, where rho:=2*cos(Pi/7) and sigma:=sin(3*Pi/7)/sin(Pi/7) = rho^2-1 are the ratios of the smaller and larger diagonal length to the side length in a regular 7-gon (heptagon). See the Steinbach reference where the basis <1,rho,sigma> is used in an extension of the rational field. (rho*sigma)^n = C(n) + B(n)*rho + A(n)*sigma,n>=0, with B(n)= |A122600(n-1)|, B(0)=0, and A(n)= A181879(n). For the nonpositive powers see A085810(n)*(-1)^n, A181880(n-2)*(-1)^n and A116423(n+1)*(-1)^(n+1), respectively. See also a comment under A052547.
We have a(n)=cs(3n+1), where the sequence cs(n) and its two conjugate sequences as(n) and bs(n) are defined in the comments to the sequence A214683 (see also A215076, A215100, A006053). We call the sequence a(n) the Ramanujan-type sequence number 5 for the argument 2Pi/7. Since as(3n+1)=bs(3n+1)=0, we obtain the following relation: 49^(1/3)*a(n) = (c(1)/c(4))^(n + 1/3) + (c(4)/c(2))^(n + 1/3) + (c(2)/c(1))^(n + 1/3), where c(j) := Cos(2Pi/7) (for more details and proofs see Witula et al.'s papers). - Roman Witula, Aug 02 2012

			a(7)=1919 because M^7= [1919,3458,4312;3458,6231,7770;4312,7770,9689].

R. Witula, E. Hetmaniok and 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.

G. C. Greubel, Table of n, a(n) for n = 1..1000
P. Steinbach, Golden fields: a case for the heptagon, Math. Mag. 70 (1997), no. 1, 22-31, MR 1439165
Roman Witula, Ramanujan Type Trigonometric Formulas: The General Form for the Argument 2*Pi/7, Journal of Integer Sequences, Vol. 12 (2009), Article 09.8.5.
Roman Witula, Full Description of Ramanujan Cubic Polynomials, Journal of Integer Sequences, Vol. 13 (2010), Article 10.5.7.
Roman Witula, Ramanujan Cubic Polynomials of the Second Kind, Journal of Integer Sequences, Vol. 13 (2010), Article 10.7.5.
Roman Witula, Ramanujan Type Trigonometric Formulae, Demonstratio Math. 45 (2012) 779-796.
Index entries for linear recurrences with constant coefficients, signature (3,4,1).

Cf. A006053, A214683, A215076, A215100.

a:=[0,2,7]; [ n le 3 select a[n] else 3*Self(n-1) + 4*Self(n-2) + Self(n-3): n in [1..25]]; // Marius A. Burtea, Oct 03 2019

with(linalg): M[1]:=matrix(3,3,[0,1,1,1,1,2,1,2,2]): for n from 2 to 25 do M[n]:=multiply(M[1],M[n-1]) od: seq(M[n][1,1],n=1..25);

LinearRecurrence[{3,4,1},{0,2,7},40] (* Roman Witula, Aug 02 2012 *)

a(n)=([0,1,0; 0,0,1; 1,4,3]^(n-1)*[0;2;7])[1,1] \\ Charles R Greathouse IV, Jun 22 2016

@CachedFunction
def a(n): # a = A120757
    if (n<3): return (0,2,7)[n]
    else: return 3*a(n-1) + 4*a(n-2) + a(n-3)
[a(n) for n in range(40)] # G. C. Greubel, Nov 25 2022

a(n) = 3*a(n-1) + 4*a(n-2) + a(n-3) (follows from the minimal polynomial of the matrix M). See also the o.g.f. given in the name.

Edited by N. J. A. Sloane, Dec 03 2006

New name, old name as comment; o.g.f.; reference.

A116423 Binomial transform of A006053.

Original entry on oeis.org

0, 1, 3, 9, 26, 74, 209, 588, 1651, 4631, 12983, 36388, 101972, 285741, 800660, 2243445, 6286059, 17613241, 49351342, 138279586, 387451077, 1085614208, 3041824015, 8523002359, 23880923183, 66912861640, 187485674652, 525323190505, 1471922876424, 4124236259529

Offset: 1

Gary W. Adamson, Feb 14 2006

nonn

a(n)/a(n-1) tends to 2.801... = 1 + 2*cos(Pi/7).
A(n) := a(n+1)*(-1)^(n+1) appears in the following formula for the nonpositive powers of rho*sigma, where rho:=2*cos(Pi/7) and sigma:=sin(3*Pi/7)/sin(Pi/7) = rho^2-1 are the ratios of the smaller and larger diagonal length to the side length in a regular 7-gon (heptagon). See the Steinbach reference where the basis <1,rho,sigma> is used in an extension of the rational field. (rho*sigma)^(-n) = C(n) + B(n)*rho + A(n)*sigma, n >= 0, with C(n)= A085810(n)*(-1)^n, and B(n)= A181880(n-2)*(-1)^n. For the nonnegative powers see A120757(n), |A122600(n-1)| and A181879(n), respectively. See also a comment under A052547.
This sequence is constructible as a spiral tiling of similar trapezoids, as follows: start with an isosceles trapezoid with side lengths 3,1,4,1. Each new trapezoid is rotated and scaled so one leg fills all unoccupied space on the short base of the previous trapezoid. a(n) is given by the length of the n-th trapezoid's legs. This process is identical to the recursion relation added by R. J. Mathar in the Formula section. See the Links section for an illustration. - Andrew B. Hudson, Jun 19 2019

			a(5) = 26 = 1*0 + 1*4 + 4*1 + 4*3 + 6*1 = 4 + 4 + 12 + 6 = 26.

Jinyuan Wang, Table of n, a(n) for n = 1..1000
Andrew B. Hudson, Illustration of the first 7 terms as a spiral tiling of similar trapezoids.
Peter Steinbach, Golden Fields: A Case for the Heptagon, Mathematics Magazine, Vol. 70, No. 1, Feb. 1997.
Index entries for linear recurrences with constant coefficients, signature (4,-3,-1)

Cf. A006053.

I:=[0,1,3]; [n le 3 select I[n] else 4*Self(n-1)-3*Self(n-2)-Self(n-3): n in [1..30]]; // Vincenzo Librandi, Jul 11 2019

LinearRecurrence[{4, -3, -1}, {0, 1, 3}, 40] (* Vincenzo Librandi, Jul 11 2019 *)

concat(0, Vec(x^2*(1-x)/(1-4*x+3*x^2+x^3) + O(x^50))) \\ Michel Marcus, Sep 13 2014

Binomial transform of A006053 starting with A006053(1): (0, 1, 1, 3, 4, 9, 14, ...).
From R. J. Mathar, Apr 02 2008: (Start)
O.g.f.: x^2(1-x)/(1 - 4x + 3x^2 + x^3).
a(n) = 4*a(n-1) - 3*a(n-2) - a(n-3). (End)

More terms from R. J. Mathar, Apr 02 2008

More terms from Michel Marcus, Sep 13 2014

A181879 Expansion of x(1+x)/(1-3x-4*x^2-x^3).

Original entry on oeis.org

0, 1, 4, 16, 65, 263, 1065, 4312, 17459, 70690, 286218, 1158873, 4692181, 18998253, 76922356, 311452261, 1261044460, 5105864780, 20673224441, 83704176903, 338911293253, 1372223811812, 5556020785351, 22495868896554, 91083913642878, 368791237300201, 1493205235368669, 6045864568949689, 24479205885623944, 99114281168039257, 401305531615563236

Offset: 0

Wolfdieter Lang, Nov 26 2010

a(n) appears in the following formula for the nonnegative powers of rho*sigma, where rho:=2*cos(Pi/7) and sigma:=sin(3*Pi/7)/sin(Pi/7)= rho^2-1 are the ratios of the smaller and larger diagonal length to the side length in a regular 7-gon (heptagon). See the Steinbach reference where the basis <1,rho,sigma> is used in an extension of the rational field, called there Q(rho). (rho*sigma)^n = C(n) + B(n)*rho + a(n)*sigma,n>=0, with C(n)= A120757(n) with C(0):=1, and B(n)= |A122600(n-1)| with B(0)=1. For the nonpositive powers see A085810(n)*(-1)^n, A181880(n-2)*(-1)^n and A116423(n+1)*(-1)^(n+1), respectively. See also a comment under A052547.

P. Steinbach, Golden fields: a case for the heptagon, Math. Mag. 70 (1997), no. 1, 22-31.
Index entries for linear recurrences with constant coefficients, signature (3,4,1).

CoefficientList[Series[x (1+x)/(1-3x-4x^2-x^3),{x,0,40}],x] (* or *) LinearRecurrence[{3,4,1},{0,1,4},40] (* Harvey P. Dale, Feb 04 2024 *)

Vec((1+x)/(1-3*x-4*x^2-x^3)+O(x^99)) \\ Charles R Greathouse IV, Sep 24 2012

a(n) = 3*a(n-1) + 4*a(n-2) + a(n-3), n>=2, a(-1):=1, a(0)=0, a(1)=1.

A181880 Expansion of 1/(1-4x-3x^2-x^3).

Original entry on oeis.org

1, 4, 19, 89, 417, 1954, 9156, 42903, 201034, 942001, 4414009, 20683073, 96916320, 454128508, 2127946065, 9971086104, 46722311119, 218930448853, 1025859814873, 4806952917170, 22524321562152, 105544004814991, 494555936863590, 2317380083461485, 10858732149251701, 50881624784254849, 238420075668235984, 1117183909174960184, 5234877488488803537, 24529481757148330684

Offset: 0

Wolfdieter Lang, Nov 27 2010

B(n):=a(n-2)*(-1)^n, B(0):=0, B(1):=0, (o.g.f. x^2/(1 + 4*x + 3*x^2 -x^3))appears in the following formula for the nonpositive powers of rho*sigma, where rho:=2*cos(Pi/7) and sigma:=sin(3*Pi/7)/sin(Pi/7) = rho^2-1 are the ratios of the smaller and larger diagonal length to the side length in a regular 7-gon (heptagon). See the Steinbach reference where the basis <1,rho,sigma> is used in an extension of the rational field. (rho*sigma)^(-n) = C(n) + B(n)*rho + A(n)*sigma,n>=0, with C(n)= A085810(n)*(-1)^n, and A(n)= A116423(n+1)*(-1)^(n+1). For the nonnegative powers see A120757(n), |A122600(n-1)| and A181879(n), respectively. See also a comment under A052547.

P. Steinbach, Golden fields: a case for the heptagon, Math. Mag. 70 (1997), no. 1, 22-31.
Index entries for linear recurrences with constant coefficients, signature (4, 3, 1).

CoefficientList[Series[1/(1-4*x-3*x^2-x^3),{x,0,40}],x] (* or *) LinearRecurrence[{4,3,1},{1,4,19},40] (* Vladimir Joseph Stephan Orlovsky, Feb 01 2012 *)

O.g.f.: 1/(1-4*x-3*x^2-x^3).

a(n) = 4*a(n) + 3*a(n-2) +a(n-3), n>=2, a(-1):=0, a(0)=1, a(1)=4.

A123218 Irregular triangle formed by coefficients of polynomials defined by P(n,k,x) = f(n,k)(2x)^k(1 - x^2)^(n - k), where f(n, k) = (-1)^floor((k + 1)/2) binomial(n - floor((k + 1)/2), floor(k/2)).

Original entry on oeis.org

1, 1, -2, -1, 1, -2, -6, 2, 1, 1, -2, -11, 12, 11, -2, -1, 1, -2, -16, 22, 46, -22, -16, 2, 1, 1, -2, -21, 32, 106, -92, -106, 32, 21, -2, -1, 1, -2, -26, 42, 191, -212, -396, 212, 191, -42, -26, 2, 1, 1, -2, -31, 52, 301, -382, -1011, 792, 1011, -382, -301, 52, 31, -2, -1

Offset: 1

Roger L. Bagula and Gary W. Adamson, Oct 04 2006

			Triangle begins with:
1;
1, -2,  -1;
1, -2,  -6,  2,  1;
1, -2, -11, 12, 11,  -2,  -1;
1, -2, -16, 22, 46, -22, -16, 2, 1;

G. C. Greubel, Rows n = 1..101 of triangle, flattened
P. Steinbach, Golden fields: a case for the heptagon, Math. Mag. 70 (1997), no. 1, 22-31.

Cf. A038223, A122600, A122753.

f[n_, k_]:= (-1)^Floor[(k+1)/2]*Binomial[n -Floor[(k+1)/2], Floor[k/2]]; Table[CoefficientList[Sum[f[n, k]*(2*x)^k*(1-x^2)^(n-k), {k, 0, n}], x], {n, 0, 10}]//Flatten

Let f(n, k) = (-1)^floor((k + 1)/2)*binomial(n - floor((k + 1)/2), floor(k/2)) then the polynomials P(n, k, x) = f(n,k)*(2*x)^k*(1 - x^2)^(n - k) for an irregular triangle of coefficients.

A321703 a(n) = 3a(n-1) + 4a(n-2) + a(n-3), a(0) = 1, a(1) = 1, a(2) = 5.

Original entry on oeis.org

1, 1, 5, 20, 81, 328, 1328, 5377, 21771, 88149, 356908, 1445091, 5851054, 23690434, 95920609, 388374617, 1572496721, 6366909240, 25779089221, 104377401344, 422615470156, 1711135105065, 6928244597163, 28051889681905, 113579782539432, 459875150943079, 1861996472668870, 7539069804318358, 30525070454573633, 123593487053663201, 500419812783602493

Offset: 0

Kai Wang, Jan 14 2019

In general, let {X,Y,Z} be the roots of the cubic equation x^3 + ax^2 + xt + 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.
Let X = (sin(4k)*sin(8k))/(sin(2k)*sin(2k)),
    Y = (sin(8k)*sin(2k))/(sin(4k)*sin(4k)),
    Z = (sin(2k)*sin(4k))/(sin(8k)*sin(8k)).
Then {X,Y,Z} are the roots of the cubic equation x^3 - 3*x^2 - 4*x - 1 = 0.
This sequence: (a, b, c) = (3, 4, 1), (u, v, w) = (1/(sqrt(7)*tan(4k)), 1/(sqrt(7)*tan(8k)), 1/(sqrt(7)*tan(2k))).
A122600: (a, b, c) = (3, 4, 1), (u, v, w) = (1/(sqrt(7)*tan(2k)), 1/(sqrt(7)*tan(4k)), 1/(sqrt(7)*tan(8k))).
A321715: (a, b, c) = (3, 4, 1), (u, v, w) = (1/(sqrt(7)*tan(8k)), 1/(sqrt(7)*tan(2k)), 1/(sqrt(7)*tan(4k))).

Colin Barker, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (3,4,1).

Cf. A321715, A122600.

CoefficientList[Series[(-1 + 2 x + 2 x^2)/(-1 + 3 x + 4 x^2 + x^3), {x, 0, 50}], x] (* Stefano Spezia, Jan 14 2019 *)

Vec((1 - 2*x - 2*x^2) / (1 - 3*x - 4*x^2 - x^3) + O(x^30)) \\ Colin Barker, Jan 15 2019

G.f.: (-1 + 2*x + 2*x^2)/(-1 + 3*x + 4*x^2 + x^3). - Stefano Spezia, Jan 14 2019

A321715 a(n) = 3a(n-1) + 4a(n-2) + a(n-3), a(0) = 1, a(1) = -1, a(2) = -1 .

Original entry on oeis.org

1, -1, -1, -6, -23, -94, -380, -1539, -6231, -25229, -102150, -413597, -1674620, -6780398, -27453271, -111156025, -450061557, -1822262042, -7378188379, -29873674862, -120956040144, -489741008259, -1982920860215, -8028682653825, -32507472410594, -131620068707297, -532918778418092, -2157744082494058, -8736527429861839, -35373477397979841, -143224285995880937

Offset: 0

Kai Wang, Jan 14 2019

In general, let {X,Y,Z} be the roots of the cubic equation x^3 + ax^2 + xt + 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.
Let X = (sin(4k)*sin(8k))/(sin(2k)*sin(2k)),
    Y = (sin(8k)*sin(2k))/(sin(4k)*sin(4k)),
    Z = (sin(2k)*sin(4k))/(sin(8k)*sin(8k)).
Then {X,Y,Z} are the roots of the cubic equation x^3 - 3*x^2 - 4*x - 1 = 0.
This sequence: (a, b, c) = (3, 4, 1), (u, v, w) = (1/(sqrt(7)*tan(8k)), 1/(sqrt(7)*tan(2k)), 1/(sqrt(7)*tan(4k))).
A122600: (a, b, c) = (3, 4, 1), (u, v, w) = (1/(sqrt(7)*tan(2k)), 1/(sqrt(7)*tan(4k)), 1/(sqrt(7)*tan(8k))).
A321703: (a, b, c) = (3, 4, 1), (u, v, w) = (1/(sqrt(7)*tan(4k)), 1/(sqrt(7)*tan(8k)), 1/(sqrt(7)*tan(2k))).

Colin Barker, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (3,4,1).

Cf. A321703, A122600.

Vec((1 - 4*x - 2*x^2) / (1 - 3*x - 4*x^2 - x^3) + O(x^30)) \\ Colin Barker, Jan 15 2019

G.f.: (1 - 4*x - 2*x^2) / (1 - 3*x - 4*x^2 - x^3). - Colin Barker, Jan 15 2019

cp's OEIS Frontend

A215076 a(n) = 3*a(n-1) + 4*a(n-2) + a(n-3) with a(0)=3, a(1)=3, a(2)=17.

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A085810 Number of three-choice paths along a corridor of height 5, starting from the lower side.

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A120757 Expansion of x^2*(2+x)/(1-3*x-4*x^2-x^3).

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A116423 Binomial transform of A006053.

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A181879 Expansion of x*(1+x)/(1-3*x-4*x^2-x^3).

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A181880 Expansion of 1/(1-4*x-3*x^2-x^3).

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A215076 a(n) = 3a(n-1) + 4a(n-2) + a(n-3) with a(0)=3, a(1)=3, a(2)=17.

A120757 Expansion of x^2(2+x)/(1-3x-4*x^2-x^3).

A181879 Expansion of x(1+x)/(1-3x-4*x^2-x^3).

A181880 Expansion of 1/(1-4x-3x^2-x^3).

A123218 Irregular triangle formed by coefficients of polynomials defined by P(n,k,x) = f(n,k)(2x)^k(1 - x^2)^(n - k), where f(n, k) = (-1)^floor((k + 1)/2) binomial(n - floor((k + 1)/2), floor(k/2)).

A321703 a(n) = 3a(n-1) + 4a(n-2) + a(n-3), a(0) = 1, a(1) = 1, a(2) = 5.

A321715 a(n) = 3a(n-1) + 4a(n-2) + a(n-3), a(0) = 1, a(1) = -1, a(2) = -1 .