A094650 -id:A094650 - cp's OEIS frontend

A094648 An accelerator sequence for Catalan's constant.

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, May 18 2004

The pair A094648 and the alternating sequence A033304 when joined form a two-sided sequence defined by the recurrence formula x(n+3) + x(n+2) - 2x(n+1) - x(n) = 0, n in Z, x(-1)=-2, x(0)=3, x(1)=-1 - for details see Witula's comments to A033304. - Roman Witula, Jul 25 2012
From Roman Witula, Aug 09 2012: (Start)
There exist two interesting subsequences b(n) and c(n) of the given above sequence x(n) defined by the following relations: b(n)=a(2^n) and c(n)=x(-2^n). These subsequences satisfy the following system of recurrence equations:
b(n+1)=b(n)^2-2*c(n), and c(n+1)=c(n)^2-2*b(n),
which easily follow from the general identity: x(n)^2=x(2*n)-2*x(-n), n in Z. We note that b(0)=-1, b(1)=5, b(2)=13, b(3)=117, c(0)=-2, c(1)=6, c(2)=26, c(3)=650. From the above system we deduce that all b(n) are odd, whereas all c(n) are even. Moreover we obtain c(n+1)-b(n+1)=(c(n)-b(n))*(b(n)+c(n)+2), which yields b(n+1)-c(n+1)=product{k=1,..,n}(b(k)+c(k)+2)=13*product{k=2,..,n}(b(k)+c(k)+2)=13^2*41*product{k=3,..,n}(b(k)+c(k)+2). It follows that b(n)-c(n) is divisible by 13^2*41 for every n=3,4,..., and after using the above system again each b(n) and c(n), for n=2,3,..., is divisible by 13. (End)
If we set W(n):=3*A077998(n)-A006054(n+1)-A006054(n), n=0,1,..., then a(n)=(W(n)^2-W(2*n))/2 and W(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 = (-1-c(1))^n + (-1-c(2))^n + (-1-c(4))^n, where c(j):=2*cos(2*Pi*j/7) - for the proof see Witula-Slota-Warzynski's paper. Moreover it follows from the comment at the top and from comments to A033304 that W(n+1)=A033304(n)=(-1)^(n+1)*x(-n-1). - Roman Witula, Aug 11 2012
The following trigonometric type identitities hold true: (1) -a(n-1)-a(n) = c(1)*c(2)^n + c(2)*c(4)^n + c(4)*c(1)^n and (2) a(n)-a(n+2) = c(4)*c(2)^(n+1) + c(1)*c(4)^(n+1) + c(2)*c(1)^(n+1), where a(-1)=-2 and c(j) is defined as above (see also the respective comment to A033304). For the proof see Remark 6 in Witula's paper. - Roman Witula, Aug 14 2012
It can be proved that A033304(n-1)*(-1)^n = (a(n)^2 - a(2*n))/2, n=1,2,... - Roman Witula, Sep 30 2012
With respect to the form of the trigonometric formulas describing a(n), we call this sequence the Berndt-type sequence number 19 for the argument 2*Pi/7. The A-numbers of other Berndt-type sequences numbers are given in below. - Roman Witula, Sep 30 2012

			We have a(17) = a(19) + 50000, a(4) + a(5) = -3, 2*a(7) + a(8) = 3, and 2*a(9) + a(10) = a(5). - _Roman Witula_, Sep 14 2012

G. C. Greubel, Table of n, a(n) for n = 0..3900
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.
Kai Wang, Fibonacci Numbers And Trigonometric Functions Outline, (2019).
Q. Wang, On generalized Lucas sequences, Contemp. Math. 531 (2010) 127-141
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 (-1,2,1).

Cf. A000032, A094649, A094650.

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

I:=[3,-1,5]; [n le 3 select I[n]  else -Self(n-1)+2*Self(n-2)+Self(n-3): n in [1..40]]; // Vincenzo Librandi, Jul 25 2015

CoefficientList[ Series[(3 + 2x - 2x^2)/(1 + x - 2x^2 - x^3), {x, 0, 33}], x] (* Robert G. Wilson v, May 24 2004 *)
a[n_] := Round[(2Sin[3Pi/14])^n + (-2Sin[Pi/14])^n + (-2Cos[Pi/7])^n]; Table[ a[n], {n, 0, 33}] (* Robert G. Wilson v, May 24 2004 *)
LinearRecurrence[{-1,2,1}, {3,-1,5}, 50] (* Roman Witula, Aug 09 2012 *)

x='x+O('x^30); Vec((3+2*x-2*x^2)/(1+x-2*x^2-x^3)) \\ G. C. Greubel, May 09 2018

G.f.: (3+2*x-2*x^2)/(1+x-2*x^2-x^3);
a(n) = (2*sin(3*Pi/14))^n+(-2*sin(Pi/14))^n+(-2*cos(Pi/7))^n.
a(p) == -1 mod(p), p prime. - Philippe Deléham, Oct 03 2009
a(n) = (2*cos(2*Pi/7))^n + (2*cos(4*Pi/7))^n + (2*cos(8*Pi/7))^n, which is equivalent to the formula given above (for analogous sums with sines see A215493 and A215494). Moreover we have a(n+3) + a(n+2) - 2a(n+1) - a(n) = 0 - for the proof see Witula-Slota's paper. - Roman Witula, Jul 24 2012
a(n) = 3*(-1)^n*A006053(n+2) +2*A078038(n-1). - R. J. Mathar, Nov 03 2020

A189234 Expansion of (5-4x-12x^2+6x^3+3x^4)/(1-x-4x^2+3x^3+3*x^4-x^5).

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, 2166063365, 4259693562

Offset: 0

L. Edson Jeffery, Apr 18 2011

(Start) Let U be the unit-primitive matrix (see [Jeffery])
U=U_(11,1)=
(0 1 0 0 0)
(1 0 1 0 0)
(0 1 0 1 0)
(0 0 1 0 1)
(0 0 0 1 1).
Then a(n)=Trace(U^n). (End)
Evidently one of a class of accelerator sequences for Catalan's constant based on traces of successive powers of a unit-primitive matrix U_(N,r) (0

Michael De Vlieger, Table of n, a(n) for n = 0..3532
A. Akbary, Q. Wang, A generalized Lucas sequence and permutation binomials, Proc. Amer. Math. Soc. 134 (2006) 15-22, sequence a(n) with l=11.
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 1 (k=5).
Index entries for linear recurrences with constant coefficients, signature (1,4,-3,-3,1).

A189235, A189236, A189237.

Unsigned version of A094650.

CoefficientList[Series[(5-4x-12x^2+6x^3+3x^4)/(1-x-4x^2+3x^3+ 3x^4-x^5),{x,0,40}],x] (* or *) LinearRecurrence[{1,4,-3,-3,1},{5,1,9,4,25},40] (* Harvey P. Dale, Jan 18 2012 *)

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^99)) \\ Charles R Greathouse IV, Sep 26 2012

G.f.: (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).
a(n)=a(n-1)+4*a(n-2)-3*a(n-3)-3*a(n-4)+a(n-5), {a(m)}={5,1,9,4,25}, m=0..4.
a(n)=Sum_{k=1..5} (x_k)^n; x_k=2*(-1)^(k-1)*cos(k*Pi/11).

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

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A094648 An accelerator sequence for Catalan's constant.

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A189234 Expansion of (5-4x-12x^2+6x^3+3x^4)/(1-x-4x^2+3x^3+3*x^4-x^5).

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A094649 An accelerator sequence for Catalan's constant.

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cp's OEIS Frontend

A094648 An accelerator sequence for Catalan's constant.

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A189234 Expansion of (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).

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A094649 An accelerator sequence for Catalan's constant.

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Mathematica

PARI

Formula

A189234 Expansion of (5-4x-12x^2+6x^3+3x^4)/(1-x-4x^2+3x^3+3*x^4-x^5).