A188048 -id:A188048 - cp's OEIS frontend

A215664 a(n) = 3*a(n-2) - a(n-3), with a(0)=3, a(1)=0, and a(2)=6.

3, 0, 6, -3, 18, -15, 57, -63, 186, -246, 621, -924, 2109, -3393, 7251, -12288, 25146, -44115, 87726, -157491, 307293, -560199, 1079370, -1987890, 3798309, -7043040, 13382817, -24927429, 47191491, -88165104, 166501902, -311686803, 587670810, -1101562311

Offset: 0

Roman Witula, Aug 20 2012

The Berndt-type sequence number 5 for the argument 2Pi/9 defined by the first relation from the section "Formula" below. The respective sums with negative powers of the cosines form the sequence A215885. Additionally if we set b(n) = c(1)*c(2)^n + c(2)*c(4)^n + c(4)*c(1)^n and c(n) = c(4)*c(2)^n + c(1)*c(4)^n + c(2)*c(1)^n, where c(j):=2*cos(2*Pi*j/9), then the following system of recurrence equations holds true: b(n) - b(n+1) = a(n), a(n+1) - a(n) = c(n+1), a(n+2) - 2*a(n)=c(n). All three sequences satisfy the same recurrence relation: X(n+3) - 3*X(n+1) + X(n) = 0. Moreover we have a(n+1) + A215665(n) + A215666(n) = 0 since c(1) + c(2) + c(4) = 0, b(n)=A215665(n) and c(n)=A215666(n).
If X(n) = 3*X(n-2) - X(n-3), n in Z, with X(n) = a(n) for every n=0,1,..., then X(-n) = A215885(n) for every n=0,1,...
From initial values and the recurrence formula we deduce that a(n)/3 and a(3n+1)/9 are all integers. We have a(n)=3*(-1)^n *A188048(n) and a(2n)=A215455(n). Furthermore the following decomposition holds: (X - c(1)^n)*(X - c(2)^n)*(X - c(4)^n) = X^3 - a(n)*X^2 + ((a(n)^2 - a(2*n))/2)*X + (-1)^(n+1), which implies the relation (c(1)*c(2))^n + (c(1)*c(4))^n + (c(2)*c(4))^n = (-c(1))^(-n) + (-c(2))^(-n) + (-c(4))^(-n) = (a(n)^2 - a(2*n))/2.

			We have c(1)^2 + c(2)^2 + c(4)^2 + 2*(c(1)^3 + c(2)^3 + c(4)^3) = 0 and 3*a(7) + a(8) = a(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).

Michael De Vlieger, Table of n, a(n) for n = 0..3649
Kai Wang, Fibonacci Numbers And Trigonometric Functions Outline, (2019).
Index entries for linear recurrences with constant coefficients, signature (0,3,-1).

Cf. A215455, A215634, A215635, A215636, A215007, A214699, A214683.

Mathematica
```
LinearRecurrence[{0,3,-1}, {3,0,6}, 50]
```

Vec(3*(1-x^2)/(1-3*x^2+x^3)+O(x^99)) \\ Charles R Greathouse IV, Sep 27 2012

a(n) = c(1)^n + c(2)^n + c(4)^n, where c(j) := 2*cos(2*Pi*j/9).

G.f.: 3*(1-x^2)/(1-3*x^2+x^3).

A094831 Number of (s(0), s(1), ..., s(2n)) such that 0 < s(i) < 9 and |s(i) - s(i-1)| = 1 for i = 1,2,...,2n, s(0) = 3, s(2n) = 3.

Original entry on oeis.org

1, 2, 6, 19, 62, 207, 703, 2417, 8382, 29242, 102431, 359790, 1266103, 4460939, 15730497, 55500634, 195890270, 691566411, 2441886670, 8623112591, 30453261927, 107553444913, 379864424726, 1341658806066, 4738726458775

Offset: 0

Herbert Kociemba, Jun 13 2004

In general, a(n) = (2/m)*Sum_{r=1..m-1} sin(r*j*Pi/m)*sin(r*k*Pi/m)*(2*cos(r*Pi/m))^(2n) counts (s(0), s(1), ..., s(2n)) such that 0 < s(i) < m and |s(i) - s(i-1)| = 1 for i = 1,2,...,2n, s(0) = j, s(2n) = k.

A comparison of their recurrence relations shows that this sequence is the even bisection of A188048. - John Blythe Dobson, Jun 20 2015

Michael De Vlieger, Table of n, a(n) for n = 0..1825
Index entries for linear recurrences with constant coefficients, signature (6,-9,1).

Cf. A188048.

CoefficientList[Series[(1 - 4 x + 3 x^2)/(1 - 6 x + 9 x^2 - x^3), {x, 0, 24}], x] (* Michael De Vlieger, Feb 12 2022 *)

Vec((1-4*x+3*x^2)/(1-6*x+9*x^2-x^3) + O(x^30)) \\ Michel Marcus, Jun 21 2015

a(n) = (2/9) * Sum_{r=1..8} sin(r*Pi/3)^2*(2*cos(r*Pi/9))^(2*n).
a(n) = 6*a(n-1) - 9*a(n-2) + a(n-3).
G.f.: (1-4*x+3*x^2)/(1-6*x+9*x^2-x^3).

A094233 Number of closed walks of length n at a vertex of the cyclic graph on 9 nodes C_9.

Original entry on oeis.org

1, 0, 2, 0, 6, 0, 20, 0, 70, 2, 252, 22, 924, 156, 3432, 910, 12870, 4760, 48622, 23256, 184796, 108528, 705894, 490314, 2708204, 2163150, 10430500, 9373652, 40313160, 40060078, 156305070, 169345560, 607812102, 709645552, 2369918628, 2952780320

Offset: 0

Herbert Kociemba, May 29 2004

In general, a(n,m) = (2^n/m)*Sum_{k=0..m-1} cos(2*Pi*k/m)^n gives the number of closed walks of length n at a vertex of the cyclic graph on m nodes C_m.

Index entries for linear recurrences with constant coefficients, signature (1,5,-4,-5,2).

Cf. A078008, A054877, A047849.

f[n_] := FullSimplify[ TrigToExp[ 2^n/9 Sum[ Cos[2Pi*k/9]^n, {k, 0, 8}]]]; Table[ f[n], {n, 0, 40}] (* Robert G. Wilson v, Jun 01 2004 *)

a(n) = (2^n/9)*Sum_{k=0..8} cos(2*Pi*k/9)^n.
G.f.: -(x-1)*(x^3+3*x^2-1)/((2*x-1)*(x+1)*(x^3-3*x^2+1)). - Maksym Voznyy (voznyy(AT)mail.ru), Aug 11 2009
9*a(n) = 2*(-1)^n +2^n +6*(-1)^n*A188048(n). - R. J. Mathar, Nov 03 2020

More terms from Robert G. Wilson v, Jun 01 2004

A094834 Number of (s(0), s(1), ..., s(2n+1)) such that 0 < s(i) < 9 and |s(i) - s(i-1)| = 1 for i = 1,2,...,2n+1, s(0) = 3, s(2n+1) = 6.

Original entry on oeis.org

1, 5, 21, 82, 308, 1131, 4096, 14705, 52497, 186733, 662630, 2347680, 8309143, 29388368, 103895601, 367187437, 1297452581, 4583924154, 16193659132, 57204089987, 202065531888, 713750040577, 2521114546457, 8905002445437

Offset: 1

Herbert Kociemba, Jun 13 2004

In general, a(n) = (2/m)*Sum_{r=1..m-1} sin(r*j*Pi/m)*sin(r*k*Pi/m)*(2*cos(r*Pi/m))^(2n+1) counts (s(0), s(1), ..., s(2n+1)) such that 0 < s(i) < m and |s(i) - s(i-1)| = 1 for i = 1,2,...,2n+1, s(0) = j, s(2n+1) = k.

This sequence is the odd bisection of A188048. - John Blythe Dobson, Jun 20 2015

Index entries for linear recurrences with constant coefficients, signature (6,-9,1).

I:=[1,5,21]; [n le 3 select I[n] else 6*Self(n-1)-9*Self(n-2)+Self(n-3): n in [1..30]]; // Vincenzo Librandi, Jun 21 2015

CoefficientList[Series[(x - 1)/(- 1 + 6 x - 9 x^2 + x^3), {x, 0, 20}], x] (* Vincenzo Librandi, Jun 21 2015 *)
LinearRecurrence[{6,-9,1},{1,5,21},30] (* Harvey P. Dale, Dec 27 2019 *)

a(n) = (2/9)*Sum_{r=1..8} sin(r*Pi/3)*sin(2*r*Pi/3)*(2*cos(r*Pi/9))^(2n+1).
a(n) = 6*a(n-1) - 9*a(n-2) + a(n-3).
G.f.: x(-1+x)/(-1 + 6x - 9x^2 + x^3).
a(n) = A094829(n+1) - A094829(n). - R. J. Mathar, Nov 15 2019

cp's OEIS Frontend

A215664 a(n) = 3*a(n-2) - a(n-3), with a(0)=3, a(1)=0, and a(2)=6.

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

Programs

Mathematica

PARI

Formula

A094831 Number of (s(0), s(1), ..., s(2n)) such that 0 < s(i) < 9 and |s(i) - s(i-1)| = 1 for i = 1,2,...,2n, s(0) = 3, s(2n) = 3.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

PARI

Formula

A094233 Number of closed walks of length n at a vertex of the cyclic graph on 9 nodes C_9.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

Formula

Extensions

A094834 Number of (s(0), s(1), ..., s(2n+1)) such that 0 < s(i) < 9 and |s(i) - s(i-1)| = 1 for i = 1,2,...,2n+1, s(0) = 3, s(2n+1) = 6.

Views

Author

Keywords

Comments

Links

Programs

Magma

Mathematica

Formula