OFFSET
0,3
COMMENTS
The Berndt-type sequence number 2 for argument 2Pi/7 is defined by the following relation: a(n) = -(2^(2n-1)/sqrt(7))*(((s(1))^2n/s(2)) + ((s(4))^2n/s(1)) + ((s(2))^2n/s(4))), where s(j) := sin(2Pi*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.
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, A. Zaharescu, Finite trigonometric sums and class numbers, Math. Ann. 330 (2004), 551-575.
B. C. Berndt, 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.
R. Wituła, P. Lorenc, M. Różański, 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).
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
EXAMPLE
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))).
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 *)
PROG
(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
(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
(Sage)
def A215008_list(prec):
P.<x> = 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
(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
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Roman Witula, Jul 31 2012
STATUS
approved