A215008 a(n) = 7*a(n-1) - 14*a(n-2) + 7*a(n-3), a(0)=0, a(1)=1, a(2)=5.

0, 1, 5, 21, 84, 329, 1274, 4900, 18767, 71687, 273371, 1041348, 3964051, 15083082, 57374296, 218205281, 829778397, 3155194917, 11996903828, 45614046737, 173428037986, 659377938380, 2506951364015, 9531364676687, 36237879209259, 137774708539300, 523812203582283, 1991504659990594

Offset: 0

Author

Roman Witula, Jul 31 2012

Keywords

• nonn
• easy

Comments

The Berndt-type sequence number 2 for argument 2*Pi/7 is defined by the following relation: a(n) = -(2^(2*n-1)/sqrt(7))*((s(1))^(2*n)/s(2) + (s(4))^(2*n)/s(1) + (s(2))^(2*n)/s(4)), where s(j) := sin(2*Pi*j/7) - see also sequence A215007. This sequence was motivated by Berndt's et al. papers.

We note that a(n) = A002054(n) for n=0,1,...,4, and A002054(5) - a(5) = 1. Moreover, we have a(n+1)=A026027(n) for n=0,...,6, and A026027(7) - a(8) = 1. The characteristic polynomial of a(n) has the form x^3 -7*x^2 +14*x -7 = (x-(2*s(1))^2)*(x-(2*s(2))^2)*(x-(2*s(4))^2) and was known to Johannes Kepler (1571-1630) - see Witula's book and Savio-Suryanarayan's paper.

Examples

			We have a(6)<a(8), but the following amazing equality holds:
  (s(1))^6/s(2) + (s(4))^6/s(1) + (s(2))^6/s(4) = (s(1))^8/s(2) + (s(4))^8/s(1) + (s(2))^8/s(4) = -21*sqrt(7)/32.
It can be also proved that
  (s(1))^3/s(2) + (s(4))^3/s(1) + (s(2))^3/s(4) = (s(1))^5/s(2) + (s(4))^5/s(1) + (s(2))^5/s(4) = (s(1))^7/s(2) + (s(4))^7/s(1) + (s(2))^7/s(4).
		

References

• R. Witula, Complex numbers, Polynomials and Fractial Partial Decompositions, T.3, Silesian Technical University Press, Gliwice 2010 (in Polish).
• R. Witula, E. Hetmaniok and 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.

Links

• G. C. Greubel, Table of n, a(n) for n = 0..1000
• B. C. Berndt and A. Zaharescu, Finite trigonometric sums and class numbers, Math. Ann. 330 (2004), 551-575.
• B. C. Berndt and L.-C. Zhang, Ramanujan's identities for eta-functions, Math. Ann. 292 (1992), 561-573.
• Z.-G. Liu, Some Eisenstein series identities related to modular equations of the seventh order, Pacific J. Math. 209 (2003), 103-130.
• D. Y. Savio and E. R. Suryanarayan, Chebyshev Polynomials and Regular Polygons, Amer. Math. Monthly, 100 (1993), 657-661.
• Roman Witula, Ramanujan Type Trigonometric Formulae, Demonstratio Math. 45 (2012) 779-796.
• Roman Wituła, P. Lorenc, M. Różański, and M. Szweda, Sums of the rational powers of roots of cubic polynomials, Zeszyty Naukowe Politechniki Slaskiej, Seria: Matematyka Stosowana z. 4, Nr. kol. 1920, 2014.
• Index entries for linear recurrences with constant coefficients, signature (7,-14,7).

Crossrefs

Cf. A215007.

Programs

GAP

a:=[0,1,5];; for n in [4..30] do a[n]:=7*(a[n-1]-2*a[n-2]+a[n-3]); od; a; # G. C. Greubel, Oct 03 2019

Magma

I:=[0,1,5]; [n le 3 select I[n] else 7*(Self(n-1) -2*Self(n-2) + Self(n-3)): n in [1..30]]; // G. C. Greubel, Feb 01 2018

Maple

seq(coeff(series(x*(1-2*x)/(1-7*x+14*x^2-7*x^3), x, n+1), x, n), n = 0..30); # G. C. Greubel, Oct 03 2019

Mathematica

LinearRecurrence[{7,-14,7},{0,1,5},30]
CoefficientList[Series[x (1-2x)/(1-7x+14x^2-7x^3),{x,0,30}],x] (* Harvey P. Dale, Jul 01 2021 *)

PARI

concat([0], Vec((x-2*x^2)/(1-7*x+14*x^2-7*x^3)+O(x^30))) \\ Charles R Greathouse IV, Sep 27 2012

Sage

def A215008_list(prec):
    P. = PowerSeriesRing(ZZ, prec)
    return P(x*(1-2*x)/(1-7*x+14*x^2-7*x^3)).list()
A215008_list(30) # G. C. Greubel, Oct 03 2019

Formula

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

a(n+1) - 2*a(n) = (1/sqrt(7))*Sum_{k=0,1,2} cot(2^k * alpha) * (2*sin(2^k * alpha))^(2n), where alpha = 2*Pi/7. - Roman Witula, May 16 2014

a(n) = A217274(n) - 2*A217274(n-1). - R. J. Mathar, Feb 05 2020