A215635 a(n) = - 12*a(n-1) - 54*a(n-2) - 112*a(n-3) - 105*a(n-4) -36*a(n-5) - 2*a(n-6), with a(0)=3, a(1)=-6, a(2)=18, a(3)=-60, a(4)=210, a(5)=-756.

3, -6, 18, -60, 210, -756, 2772, -10296, 38610, -145860, 554268, -2116296, 8112462, -31201644, 120347532, -465328200, 1803025410, -6999149124, 27213719148, -105960069864, 413078158350, -1612098272460, 6297409350492, -24620247483624, 96324799842498, -377102656201956, 1477141800784668

Offset: 0

Author

Roman Witula, Aug 18 2012

Keywords

• sign
• easy

Comments

The Berndt-type sequence number 3 for the argument 2*Pi/9 defined by the relation: X(n) = a(n) + b(n)*sqrt(2), where X(n) := ((cos(Pi/24))^(2*n) + (cos(7*Pi/24))^(2*n) + (cos(3*Pi/8))^(2*n))*(-4)^n. We have b(n) = A215636(n).

We note that above formula is the Binet form of the following recurrence sequence: X(n+3) + 6*X(n+2) + 9*X(n+1) + (2 + sqrt(2))*X(n) = 0, which is a special type of the sequence X(n)=X(n;g) defined in the comments to A215634 for g:=Pi/24. The sequences a(n) and b(n) satisfy the following system of recurrence equations: a(n) = -b(n+3)-6*b(n+2)-9*b(n+1)-2*b(n), 2*b(n) = -a(n+3)-6*a(n+2)-9*a(n+1)-2*a(n).

There exists an amazing relation: (-1)^n*a(n)=3*A000984(n) for every n=0,1,...,11 and 3*A000984(12)-a(12)=6.

Links

• Roman Witula and D. Slota, On modified Chebyshev polynomials, J. Math. Anal. Appl., 324 (2006), 321-343.
• Index entries for linear recurrences with constant coefficients, signature (-12,-54,-112,-105,-36,-2).

Crossrefs

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

Programs

Mathematica

LinearRecurrence[{-12,-54,-112,-105,-36,-2}, {3,-6,18,-60,210,-756}, 50]

PARI

Vec((3+30*x+108*x^2+168*x^3+105*x^4+18*x^5) /(1+12*x+54*x^2+112*x^3+105*x^4+36*x^5+2*x^6)+O(x^99)) \\ Charles R Greathouse IV, Oct 01 2012

Formula

G.f.: (3+30*x+108*x^2+168*x^3+105*x^4+18*x^5) / (1+12*x+54*x^2+112*x^3+105*x^4+36*x^5+2*x^6).