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).

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

Author

L. Edson Jeffery, Apr 18 2011

Keywords

• nonn
• easy

Comments

(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

Links

• 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).

Crossrefs

A189235, A189236, A189237.

Unsigned version of A094650.

Programs

Mathematica

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 *)

PARI

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

Formula

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).

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