A094648 -id:A094648 - cp's OEIS frontend

A033304 Expansion of (2 + 2x - 3x^2) / (1 - 2*x - x^2 + x^3).

2, 6, 11, 26, 57, 129, 289, 650, 1460, 3281, 7372, 16565, 37221, 83635, 187926, 422266, 948823, 2131986, 4790529, 10764221, 24186985, 54347662, 122118088, 274396853, 616564132, 1385407029, 3112981337, 6994805571, 15717185450

Offset: 0

N. J. A. Sloane

From L. Edson Jeffery, Mar 22 2011: (Start)
Let A be the unit-primitive matrix (see [Jeffery])
A=A_(7,2)=
(0 0 1)
(0 1 1)
(1 1 1).
Let B={b(n)} be this sequence shifted to the right one place and setting b(0)=3. Then B=(3,2,6,11,26,...) with generating function (3-4*x-x^2)/(1-2*x-x^2+x^3) and b(n)=Trace(A^n). (End)
The following identity hold true (a(n)^2 - a(2n+2))/2 = A094648(n+1) = (-1)^(n+1)*A096975(n+1) - for the proof see Witula et al.'s papers - Roman Witula, Jul 25 2012
We note that the joined sequences (-1)^(n+1)*a(n) and A094648(n) form a two-sided sequence defined either by the recurrence formula x(n+3) + x(n+2) - 2x(n+1) - x(n) = 0, n in Z, x(0)=3, x(-1)=-2, x(1)=-1, or by the following trigonometric identities: x(n) = (c(1))^n + (c(2))^n + (c(4))^n = (c(1)c(2))^(-n) + (c(1)c(4))^(-n) + (c(2)c(4))^(-n) = (s(2)/s(1))^n + (s(4)/s(2))^n + (s(1)/s(4))^n, for n in Z, where c(j) := 2*cos(2Pi*j/7) and s(j) := sin(2*Pi*j/7) - for the proof see Witula's and Witula et al.'s papers. - Roman Witula, Jul 25 2012
We have 4*a(n+2) - a(n) = 7*A077998(n+2). - Roman Witula, Aug 13 2012
Two very intriguing identities of trigonometric nature hold: (-1)^n*(a(n)-a(n-1)) = c(1)*c(2)^(-n) + c(2)*c(4)^(-n) + c(4)*c(1)^(-n), and (-1)^(n+1)*(a(n-1)-a(n+1)) = c(1)*c(4)^(-n-1) + c(2)*c(1)^(-n-1) + c(4)*c(2)^(-n-1), where a(-1):=3 and c(j) is defined as above. For the proof see Remark 6 in the first Witula's paper. - Roman Witula, Aug 14 2012
With respect to the form of the trigonometric formulas describing a(n), we call this sequence the Berndt-type sequence number 20 for the argument 2Pi/7. The A-numbers of other Berndt-type sequences numbers are given in below. - Roman Witula, Sep 30 2012

R. P. Stanley, Enumerative Combinatorics I, p. 244, Eq. (36).

G. C. Greubel, Table of n, a(n) for n = 0..1000
Mark W. Coffey, James L. Hindmarsh, Matthew C. Lettington, John Pryce, On Higher Dimensional Interlacing Fibonacci Sequences, Continued Fractions and Chebyshev Polynomials, arXiv:1502.03085 [math.NT], 2015 (see p. 40).
L. E. Jeffery, Unit-primitive matrices
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, 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 (2,1,-1).

Cf. A066170, A006356, A096975.

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

I:=[2,6,11]; [n le 3 select I[n] else 2*Self(n-1) +Self(n-2) - Self(n-3): n in [1..30]]; // G. C. Greubel, Apr 19 2018

CoefficientList[Series[(2+2x-3x^2)/(1-2x-x^2+x^3),{x,0,50}], x]  (* Harvey P. Dale, Mar 14 2011 *)
LinearRecurrence[{2, 1, -1}, {2, 6, 11}, 29] (* Jean-François Alcover, Sep 27 2017 *)

{a(n)=if(n<0, n=-n; polsym(x^3-x^2-2*x+1,n-1)[n], n+=2; polsym(1-x-2*x^2+x^3,n-1)[n])} /* Michael Somos, Aug 03 2006 */

x='x+O('x^99); Vec((2+2*x-3*x^2)/(1-2*x-x^2+x^3)) \\ Altug Alkan, Apr 19 2018

a(-1-n) = A096975(n).
a(n) = (1-2*cos(1/7*Pi))^(n+1)+(1+2*cos(2/7*Pi))^(n+1)+(1-2*cos(3/7*Pi))^(n+1). - Vladeta Jovovic, Jun 27 2001
a(n) = trace of (n+1)-th power of the 3 X 3 matrix (in the example of A066170): [1 1 1 / 1 1 0 / 1 0 0]. Alternatively, the sum of the (n+1)st powers of the roots of the corresponding characteristic polynomial: x^3 - 2*x^2 - x + 1 = 0. a(n) = A006356(n) + A006356(n-1) + 2*A006356(n-2). E.g., a(3) = 26 = the trace of M^4. The characteristic polynomial of this matrix (see A066170) is x^3 - 2*x^2 - x + 1 and the roots are 2.24697960372..., -0.8019377358... and 0.55495813208... = a, b, c. Then Sum(a^4 + b^4 + c^4) = 26. - Gary W. Adamson, Feb 01 2004
(-1)^(n+1)*a(n) = (c(1))^(-n-1) + (c(2))^(-n-1) + (c(3))^(-n-1) = (c(1)c(2))^(n+1) + (c(1)c(4))^(n+1) + (c(2)c(4))^(n+1) = (s(1)/s(2))^(n+1) + (s(2)/s(4))^(n+1) + (s(4)/s(1))^(n+1), where c(j) := 2*cos(2*Pi*j/7) and s(j) := sin(2*Pi*j/7) - for the proof see Witula's and Witula et al.'s papers. - Roman Witula, Jul 25 2012
a(n) = 3*A077998(n+1) - A006054(n+2) - A006054(n+1). - Roman Witula, Aug 13 2012
a(n)*(-1)^(n+1) = (A094648(n+1)^2 - A094648(2*(n+1)))/2. - Roman Witula, Sep 30 2012

A217274 a(n) = 7a(n-1) - 14a(n-2) + 7*a(n-3) with a(0)=0, a(1)=1, a(2)=7.

Original entry on oeis.org

0, 1, 7, 35, 154, 637, 2548, 9996, 38759, 149205, 571781, 2184910, 8333871, 31750824, 120875944, 459957169, 1749692735, 6654580387, 25306064602, 96226175941, 365880389868, 1391138718116, 5289228800247, 20109822277181, 76457523763621, 290689756066542

Offset: 0

Roman Witula, Sep 29 2012

This is the Berndt-type sequence number 18 for the argument 2*Pi/7 defined by the relation
a(n)*sqrt(7) = c(4)*s(1)^(2n+1) + c(2)*s(4)^(2n+1) + c(1)*s(2)^(2n+1) = (1/s(4))*s(1)^(2n+2) + (1/s(2))*s(4)^(2n+2) + (1/s(1))*s(2)^(2n+2), where c(j) := 2*cos(2*Pi*j/7) and s(j) := 2*sin(2*Pi*j/7) (for the sums of the respective even powers see A094429). For the proof of this formula see the Witula/Slota and Witula references.
The definitions of the other Berndt-type sequences for the argument 2*Pi/7 (with numbers from 1 to 17) are in the cross references.
We note that all numbers of the form a(n)*7^(-floor((n+1)/3)) = A217444(n) are integers.
It can be proved that Sum_{k=2..n}a(k) = 7*(a(n-1) - a(n-2)).

			Writing c(j) as cj and s(k) as sk,
we have 7*sqrt(7) = c4*s1^5 + c2*s4^5 + c1*s2^5
and c4*s1^13 + c2*s4^13 + c1*s2^13 = 4(c4*s1^11 + c2*s4^11 + c1*s2^11).
We note that a(9) = 87*a(3)*a(2)^2 and a(11) = 2*a(3)*a(5)*a(2)^2.

G. C. Greubel, Table of n, a(n) for n = 0..1000
R. Witula, Ramanujan type trigonometric formulas, Demonstratio Math., Vol. XLV, No. 4, 2012, pp. 789-796.
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).

Cf. A033304, A094429, A094430, A094648, A108716, A215007, A215008, A215143, A215493, A215494, A215510, A215512, A215575, A215694, A215695, A215794, A215817, A215828, A215877, A217444.

I:=[0,1,7]; [n le 3 select I[n] else 7*Self(n-1)-14*Self(n-2)+7*Self(n-3): n in [1..30]]; // Vincenzo Librandi, Jul 26 2015

LinearRecurrence[{7,-14,7}, {0,1,7}, 30]
CoefficientList[Series[x/(1 - 7*x + 14*x^2 - 7*x^3), {x,0,50}], x] (* G. C. Greubel, Apr 16 2017 *)

a[0]:0$
a[1]:1$
a[2]:7$
a[n]:=7*a[n-1] - 14*a[n-2] + 7*a[n-3];
makelist(a[n], n, 0, 25); /* Martin Ettl, Oct 11 2012 */

concat(0, Vec(x/(1-7*x+14*x^2-7*x^3) + O(x^40))) \\ Michel Marcus, Jul 25 2015

G.f.: x/(1-7*x+14*x^2-7*x^3).

A274032 Sum of n-th powers of the roots of x^3 + 9*x^2 - x - 1.

Original entry on oeis.org

3, -9, 83, -753, 6851, -62329, 567059, -5159009, 46935811, -427014249, 3884905043, -35344223825, 321555905219, -2925462465753, 26615373873171, -242142271419073, 2202970354179075, -20042260085157577, 182341168849178195, -1658909809373582257

Offset: 0

Kai Wang, Jun 07 2016

A Berndt-type sequence for tan(2*Pi/7).
a(n) is always an integer.
a(n) is x1^n + x2^n + x3^n, where x1, x2, x3 are the roots of the polynomial
x^3 + 9*x^2 - x - 1.
x1 = tan(Pi/7)/tan(2*Pi/7),
x2 = tan(2*Pi/7)/tan(4*Pi/7),
x3 = tan(4*Pi/7)/tan(Pi/7).
This is a two sided sequence. The other half is A274075. - Kai Wang, Aug 02 2016

Seiichi Manyama, Table of n, a(n) for n = 0..1000
B. C. Berndt, L.-C. Zhang, Ramanujan's identities for eta-functions, Math. Ann. 292 (1992), 561-573.
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 (-9,1,1).

Cf. A033304, A094648, A215076, A274075.

Vec((3+18*x-x^2)/(1+9*x-x^2-x^3) + O(x^30)) \\ Colin Barker, Jun 07 2016

polsym(x^3 + 9*x^2 - x - 1, 30) \\ Charles R Greathouse IV, Jul 20 2016

a(n) = (tan(Pi/7)/tan(2*Pi/7))^n + (-tan(2*Pi/7)/tan(3*Pi/7))^n + (-tan(3*Pi/7)/tan(Pi/7))^n.
From Colin Barker, Jun 07 2016: (Start)
a(n) = -9*a(n-1)+a(n-2)+a(n-3) for n>2.
G.f.: (3+18*x-x^2) / (1+9*x-x^2-x^3).
(End)

Edited by N. J. A. Sloane, Jun 07 2016

A274075 Sum of n-th powers of the roots of x^3 + x^2 - 9*x - 1.

Original entry on oeis.org

3, -1, 19, -25, 195, -401, 2131, -5545, 24323, -72097, 285459, -910009, 3407043, -11311665, 41065043, -139462985, 497736707, -1711838529, 6052005907, -20960815961, 73717030595, -256312368337, 898804827731, -3131899112169, 10964830193411, -38253117375201

Offset: 0

Kai Wang, Jun 09 2016

a(n) is always an integer.
This is the other half of A274032.
a(n) is x1^n + x2^n + x3^n, where x1, x2, x3 are the roots of the polynomial
x^3 + x^2 - 9*x - 1.
x1 = tan(Pi/7)/tan(4*Pi/7),
x2 = tan(4*Pi/7)/tan(2*Pi/7),
x3 = tan(2*Pi/7)/tan(Pi/7).

Colin Barker, Table of n, a(n) for n = 0..1000
B. C. Berndt, L.-C. Zhang, Ramanujan's identities for eta-functions, Math. Ann. 292 (1992), 561-573.
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 (-1,9,1).

Cf. A033304, A094648, A215076, A274032.

FullSimplify[Table[(Tan[Pi/7]/Tan[4*Pi/7])^n + (Tan[4*Pi/7]/Tan[2*Pi/7])^n + (Tan[2*Pi/7]/Tan[Pi/7])^n, {n, 0, 12}]] (* Wesley Ivan Hurt, Jun 11 2016 *)

Vec((3+2*x-9*x^2)/(1+x-9*x^2-x^3) + O(x^30)) \\ Colin Barker, Jun 11 2016

polsym(x^3 + x^2 - 9*x - 1, 30) \\ Charles R Greathouse IV, Jul 20 2016

a(n) = (tan(Pi/7)/tan(4*Pi/7))^n + (tan(4*Pi/7)/tan(2*Pi/7))^n + (tan(2*Pi/7)/tan(Pi/7))^n.
a(n) = -a(n-1) + 9*a(n-2) + a(n-3) for n>2.
G.f.: (3+2*x-9*x^2) / (1+x-9*x^2-x^3). - Colin Barker, Jun 11 2016

A094649 An accelerator sequence for Catalan's constant.

Original entry on oeis.org

4, 1, 7, 4, 19, 16, 58, 64, 187, 247, 622, 925, 2110, 3394, 7252, 12289, 25147, 44116, 87727, 157492, 307294, 560200, 1079371, 1987891, 3798310, 7043041, 13382818, 24927430, 47191492, 88165105, 166501903, 311686804, 587670811, 1101562312

Offset: 0

Paul Barry, May 18 2004

From L. Edson Jeffery, Apr 03 2011: (Start)
Let U be the unit-primitive matrix (see [Jeffery])
U = U_(9,1) =
(0 1 0 0)
(1 0 1 0)
(0 1 0 1)
(0 0 1 1).
Then a(n) = Trace(U^n). (End)
a(n)==1 (mod 3), a(3*n+1)==1 (mod 9). - Roman Witula, Sep 14 2012

			We have a(0)+a(3)=a(1)+a(2)=8, a(3)+a(4)=a(2)+a(5)=23, and a(7)+a(8)=a(9)+a(3)=247. - _Roman Witula_, Sep 14 2012

Michael De Vlieger, Table of n, a(n) for n = 0..3649
A. Akbary and Q. Wang, On some permutation polynomials over finite fields, International Journal of Mathematics and Mathematical Sciences, 2005:16 (2005) 2631-2640.
A. Akbary and Q. Wang, A generalized Lucas sequence and permutation binomials, Proceeding of the American Mathematical Society, 134 (1) (2006), 15-22, sequence a(n) with l=9.
David M. Bradley, A Class of Series Acceleration Formulae for Catalan's Constant, The Ramanujan Journal, Vol. 3, Issue 2, 1999, pp. 159-173.
David M. Bradley, A Class of Series Acceleration Formulae for Catalan's Constant, arXiv:0706.0356 [math.CA], 2007.
Russell A. Gordon, Lucas Type Sequences and Sums of Binomial Coefficients, Integers (2023) Vol 23, Art. No. A84. See p. 21.
L. E. Jeffery, Unit-primitive matrices
Genki Shibukawa, New identities for some symmetric polynomials and their applications, arXiv:1907.00334 [math.CA], 2019.
Q. Wang, On generalized Lucas sequences, Contemp. Math. 531 (2010) 127-141, Table 2 (k=4)
Index entries for linear recurrences with constant coefficients, signature (1,3,-2,-1).

Cf. A000032, A094648, A094650.

LinearRecurrence[{1, 3, -2, -1}, {4, 1, 7, 4}, 34] (* Jean-François Alcover, Sep 21 2017 *)

Vec((4-3*x-6*x^2+2*x^3)/(1-x-3*x^2+2*x^3+x^4)+O(x^66)) /* Joerg Arndt, Apr 08 2011 */

G.f.: ( 4-3*x-6*x^2+2*x^3 ) / ( (x-1)*(x^3+3*x^2-1) )
a(n) = 1+(2*cos(Pi/9))^n+(-2*sin(Pi/18))^n+(-2*cos(2*Pi/9))^n.
a(n) = 2^n*Sum_{k=1..4} cos((2*k-1)*Pi/9)^n. - L. Edson Jeffery, Apr 03 2011
a(n) = 1 + (-1)^n*A215664(n), which is compatible with the last two formulas above. - Roman Witula, Sep 14 2012
a(n) = 3*a(n-2) + a(n-3) - 3, with a(0)=4, a(1)=1, and a(2)=7. - Roman Witula, Sep 14 2012

A096975 Trace sequence of a path graph plus loop.

Original entry on oeis.org

3, 1, 5, 4, 13, 16, 38, 57, 117, 193, 370, 639, 1186, 2094, 3827, 6829, 12389, 22220, 40169, 72220, 130338, 234609, 423065, 761945, 1373466, 2474291, 4459278, 8034394, 14478659, 26088169, 47011093, 84708772, 152642789, 275049240

Offset: 0

Paul Barry, Jul 16 2004

Let A be the adjacency matrix of the graph P_3 with a loop added at the end. Then a(n) = trace(A^n). A is a 'reverse Jordan matrix' [0,0,1;0,1,1;1,1,0]. a(n) = abs(A094648(n)).
From L. Edson Jeffery, Mar 22 2011: (Start)
Let A be the unit-primitive matrix (see [Jeffery])
A = A_(7,1) =
(0 1 0)
(1 0 1)
(0 1 1).
Then a(n) = Trace(A^n). (End)

Michael De Vlieger, Table of n, a(n) for n = 0..3910
A. Akbary, Q. Wang, A generalized Lucas sequence and permutations binomials, Proc. Am. Math. Soc. 134 (2006) 15-22, sequence a(n) with l=7.
Robin Chapman and Nicholas C. Singer, Eigenvalues of a bidiagonal matrix, Amer. Math. Monthly, 111 (2004), p. 441.
Tomislav Došlić, Mate Puljiz, Stjepan Šebek, and Josip Žubrinić, On a variant of Flory model, arXiv:2210.12411 [math.CO], 2022.
L. E. Jeffery, Unit-primitive matrix
Genki Shibukawa, New identities for some symmetric polynomials and their applications, arXiv:1907.00334 [math.CA], 2019.
Q. Wang, On generalized Lucas sequences, Contemp. Math. 531 (2010) 127-141, Table 1 (k=3).
Index entries for linear recurrences with constant coefficients, signature (1,2,-1).

Cf. A006053, A052547, A096976.

A033304(n) = a(-1-n). - Michael Somos, Aug 03 2006

CoefficientList[Series[(3 - 2 x - 2 x^2)/(1 - x - 2 x^2 + x^3), {x, 0, 33}], x] (* Michael De Vlieger, Aug 21 2019 *)

{a(n)=if(n>=0, n+=1; polsym(x^3-x^2-2*x+1,n-1)[n], n=1-n; polsym(1-x-2*x^2+x^3,n-1)[n])} /* Michael Somos, Aug 03 2006 */

a(n)=trace([0,1,0;1,0,1;0,1,1]^n); /* Joerg Arndt, Apr 30 2011 */

G.f.: (3-2*x-2*x^2)/(1-x-2*x^2+x^3);
a(n) = a(n-1) + 2*a(n-2) - a(n-3);
a(n) = (2*sqrt(7)*sin(atan(sqrt(3)/9)/3)/3+1/3)^n + (1/3-2*sqrt(7)*sin(atan(sqrt(3)/9)/3+Pi/3)/3)^n + (2*sqrt(7)*cos(acot(-sqrt(3)/9)/3)/3+1/3)^n.
a(n) = 2^n*((cos(Pi/7))^n+(cos(3*Pi/7))^n+(cos(5*Pi/7))^n). - Vladimir Shevelev, Aug 25 2010
a(n) = (-1)^n*A094648(n). - R. J. Mathar, Nov 05 2024

A274220 a(n) = (-cos(Pi/7)/cos(2Pi/7))^n + (-cos(2Pi/7)/cos(3Pi/7))^n + (cos(3Pi/7)/cos(Pi/7))^n.

Original entry on oeis.org

3, -4, 10, -25, 66, -179, 493, -1369, 3818, -10672, 29865, -83626, 234237, -656205, 1838483, -5151080, 14432666, -40438941, 113306686, -317477255, 889550021, -2492461633, 6983719214, -19567941936, 54828148469, -153625048854, 430447808073, -1206087937261, 3379383275971, -9468821484028

Offset: 0

Kai Wang, Jun 14 2016

a(n) is an integer.
This is other half of A215076.
a(n) is the sum of n-th powers of the roots of x^3 + 4*x^2 + 3*x - 1. - Greg Dresden, Mar 11 2020

			a(0) = 3, a(1) = -4, a(2) = 10, a(3) = -25.

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.

Colin Barker, Table of n, a(n) for n = 0..1000
B. C. Berndt, L.-C. Zhang, Ramanujan's identities for eta-functions, Math. Ann. 292 (1992), 561-573.
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, 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.
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, 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. A215076, A033304, A094648, A274032.

CoefficientList[Series[(3 + 8 x + 3 x^2)/(1 + 4 x + 3 x^2 - x^3), {x, 0, 29}], x] (* Michael De Vlieger, Jun 14 2016 *)

Vec((3+8*x+3*x^2)/(1+4*x+3*x^2-x^3) + O(x^30)) \\ Colin Barker, Jun 14 2016

polsym(x^3 + 4*x^2 + 3*x - 1,33) \\ Joerg Arndt, Mar 12 2020

a(n) = -4*a(n-1)-3*a(n-2)+a(n-3).

G.f.: (3+8*x+3*x^2) / (1+4*x+3*x^2-x^3). - Colin Barker, Jun 14 2016

Many terms corrected by Colin Barker, Jun 14 2016

A094650 An accelerator sequence for Catalan's constant.

Original entry on oeis.org

5, -1, 9, -4, 25, -16, 78, -64, 257, -256, 874, -1013, 3034, -3953, 10684, -15229, 38017, -58056, 136338, -219508, 491870, -824737, 1782735, -3083887, 6484514, -11489516, 23652443, -42688039, 86459608, -158270401, 316576903, -585868009, 1160673633

Offset: 0

Paul Barry, May 18 2004

A. Akbary and Q. Wang, On some permutation polynomials over finite fields, International Journal of Mathematics and Mathematical Sciences, 2005:16 (2005) 2631-2640.
A. Akbary and Q. Wang, A generalized Lucas sequence and permutation binomials, Proceeding of the American Mathematical Society, 134 (1) (2006), 15-22.
David M. Bradley, A Class of Series Acceleration Formulae for Catalan's Constant, The Ramanujan Journal, Vol. 3, Issue 2, 1999, pp. 159-173
David M. Bradley, A Class of Series Acceleration Formulae for Catalan's Constant, arXiv:0706.0356 [math.CA], 2007.
Russell A. Gordon, Lucas Type Sequences and Sums of Binomial Coefficients, Integers (2023) Vol 23, Art. No. A84. See p. 21.
Index entries for linear recurrences with constant coefficients, signature (-1,4,3,-3,-1).

Cf. A000032, A094648, A094649.

LinearRecurrence[{-1, 4, 3, -3, -1}, {5, -1, 9, -4, 25}, 33] (* Jean-François Alcover, Sep 21 2017 *)

Vec((5+4*x-12*x^2-6*x^3+3*x^4)/(1+x-4*x^2-3*x^3+3*x^4+x^5) + O(x^40)) \\ Michel Marcus, Jul 25 2015

G.f.: (5+4x-12x^2-6x^3+3x^4)/(1+x-4x^2-3x^3+3x^4+x^5).

a(n) = (2*cos(2*Pi/11))^n + (-2*cos(Pi/11))^n + (-2*sin(5*Pi/22))^n +(2*sin(3*Pi/22))^n + (-2*sin(Pi/22))^n.

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

A094659 Number of closed walks of length n at a vertex of the cyclic graph on 7 nodes C_7.

Original entry on oeis.org

1, 0, 2, 0, 6, 0, 20, 2, 70, 18, 252, 110, 924, 572, 3434, 2730, 12902, 12376, 48926, 54264, 187036, 232562, 720062, 980674, 2789164, 4086550, 10861060, 16878420, 42484682, 69242082, 166823430, 282580872, 657178982, 1148548016, 2595874468

Offset: 0

Herbert Kociemba, Jun 06 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.

Harvey P. Dale, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (1,4,-3,-2).

Cf. A199572 (m=2), A078008 (m=3), A199573 (m=4), A054877 (m=5), A047849 (bisection of m=6), A063376 (bisection of m=8), A094233 (m=9), A095929 (bisection of m=10), A087433 (bisection of m=12).

f[n_] := FullSimplify[ TrigToExp[ 2^n/7 Sum[Cos[2Pi*k/7]^n, {k, 0, 6}]]]; Table[ f[n], {n, 0, 36}] (* Robert G. Wilson v, Jun 09 2004 *)
LinearRecurrence[{1,4,-3,-2},{1,0,2,0},40] (* Harvey P. Dale, Jun 12 2014 *)

a(n) = (2^n/7)*Sum_{k=0..6} cos(2*Pi*k/7)^n.
a(n) = 7(a(n-2) - 2a(n-4) + a(n-6)) + 2a(n-7).
G.f.: (1-x-2x^2+x^3)/((2x-1)(-1-x+2x^2+x^3)).
a(0)=1, a(1)=0, a(2)=2, a(3)=0, a(n)=a(n-1)+4*a(n-2)-3*a(n-3)-2*a(n-4). - Harvey P. Dale, Jun 12 2014
7*a(n) = 2^n + 2*A094648(n). - R. J. Mathar, Nov 03 2020

More terms from Robert G. Wilson v, Jun 09 2004

cp's OEIS Frontend

A033304 Expansion of (2 + 2*x - 3*x^2) / (1 - 2*x - x^2 + x^3).

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A217274 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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A274032 Sum of n-th powers of the roots of x^3 + 9*x^2 - x - 1.

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A274075 Sum of n-th powers of the roots of x^3 + x^2 - 9*x - 1.

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A094649 An accelerator sequence for Catalan's constant.

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A096975 Trace sequence of a path graph plus loop.

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A274220 a(n) = (-cos(Pi/7)/cos(2*Pi/7))^n + (-cos(2*Pi/7)/cos(3*Pi/7))^n + (cos(3*Pi/7)/cos(Pi/7))^n.

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A033304 Expansion of (2 + 2x - 3x^2) / (1 - 2*x - x^2 + x^3).

A217274 a(n) = 7a(n-1) - 14a(n-2) + 7*a(n-3) with a(0)=0, a(1)=1, a(2)=7.

A274220 a(n) = (-cos(Pi/7)/cos(2Pi/7))^n + (-cos(2Pi/7)/cos(3Pi/7))^n + (cos(3Pi/7)/cos(Pi/7))^n.

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.