A215007 a(n) = 7*a(n-1) - 14*a(n-2) + 7*a(n-3), a(0)=1, a(1)=3, a(2)=9.

1, 3, 9, 28, 91, 308, 1078, 3871, 14161, 52479, 196196, 737793, 2785160, 10540390, 39955041, 151615947, 575723785, 2187128524, 8311078307, 31587815308, 120069510526, 456434707519, 1735184512425, 6596692255391, 25079305566420

Offset: 0

Author

Roman Witula, Jul 31 2012

Keywords

• nonn
• easy

Comments

The sequence {a(n)} we shall call the Berndt-type sequence of type 1 for the argument 2*Pi/7; our motivation comes from Berndt's et al. and my papers (see the first formula below, which is in agreement with the respective identities discussed in these papers).

We note that a(n) = A105849(n) for n=0,1,...,5, and A105849(6) - a(6) = 1. Moreover we have a(n) = 2*A215008(n) - A215008(n+1).

References

• 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.
• Roman Witula, Ramanujan Type Trigonometric Formulae, Demonstratio Math. 45 (2012) 779-796.
• R. 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. A122068, A215008.

Programs

GAP

a:=[1,3,9];; 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

R:=PowerSeriesRing(Rationals(), 30); Coefficients(R!((1-4*x+2*x^2)/(1-7*x+14*x^2-7*x^3))); // G. C. Greubel, Feb 01 2018

Maple

seq(coeff(series((1-4*x+2*x^2)/(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}, {1,3,9}, 30] (* G. C. Greubel, Feb 01 2018 *)

PARI

Vec((1-4*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 A215007_list(prec):
    P. = PowerSeriesRing(ZZ, prec)
    return P((1-4*x+2*x^2)/(1-7*x+14*x^2-7*x^3)).list()
A215007_list(30) # G. C. Greubel, Oct 03 2019

Formula

a(n) = (1/sqrt(7))*(cot(8*Pi/7)*(s(1))^2n + cot(4*Pi/7)*(s(4))^2n + cot(2*Pi/7)*(s(2))^2n), where s(j) := 2*sin(2Pi*j/7).

G.f.: (1-4*x+2*x^2)/(1-7*x+14*x^2-7*x^3).