OFFSET

0,2

COMMENTS

The Ramanujan-type sequence number 7 for the argument 2Pi/7 (see also A214683, A215112, A006053, A006054, A215076, A215100, A120757, A215560 for the numbers: 1, 1a, 2, 2a, 3-6 respectively). The sequence a(n) is one of the three special sequences (the remaining two are A215560 and A215572) connected with the following recurrence relation:

(c(1)^4/c(2))^(n/3) + (c(2)^4/c(4))^(n/3) + (c(4)^4/c(1))^(n/3) = at(n) + bt(n)*7^(1/3) + ct(n)*49^(1/3), where c(j):=2*cos(2*Pi*j/7), and the sequences at(n), bt(n), and ct(n) satisfy the following system of recurrence equations: at(n)=7*bt(n-2)+at(n-3),

bt(n)=7*ct(n-2)+bt(n-3), ct(n)=at(n-2)+ct(n-3), with at(0)=3, at(1)=at(2)=bt(0)=bt(1)=bt(2)=ct(0)=ct(1)=0, ct(2)=2 - for details see the Witula's first paper (see also A215560). It follows that a(n)=bt(3*n+1), at(3*n+1)=ct(3*n+1)=0, which implies the first formula below.

We note that all numbers a(n) are divided by 7.

REFERENCES

R. Witula, E. Hetmaniok, 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

Roman Witula, Ramanujan Type Trigonometric Formulas: The General Form for the Argument 2*Pi/7, Journal of Integer Sequences, Vol. 12 (2009), Article 09.8.5.

Roman Witula, Full Description of Ramanujan Cubic Polynomials, Journal of Integer Sequences, Vol. 13 (2010), Article 10.5.7.

Roman Witula, Ramanujan Cubic Polynomials of the Second Kind, Journal of Integer Sequences, Vol. 13 (2010), Article 10.7.5.

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

Index entries for linear recurrences with constant coefficients, signature (3,46,1).

FORMULA

7^(1/3)*a(n) = (c(1)^4/c(2))^(n+1/3) + (c(2)^4/c(4))^(n+1/3) + (c(4)^4/c(1))^(n+1/3) = (c(1)*(c(1)/c(2))^(1/3))^(3*n+1) + (c(2)*(c(2)/c(4))^(1/3))^(3*n+1) + (c(4)*(c(4)/c(1))^(1/3))^(3*n+1).

G.f.: (14*x+7*x^2)/(1-3*x-46*x^2-x^3).

EXAMPLE

We have (c(1)^4/c(2))^(4/3) + (c(2)^4/c(4))^(4/3) + (c(4)^4/c(1))^(4/3) = (2/7)*(c(1)^4/c(2))^(7/3) + (c(2)^4/c(4))^(7/3) + (c(4)^4/c(1))^(7/3)).

MATHEMATICA

LinearRecurrence[{3, 46, 1}, {0, 14, 49}, 30] (* Harvey P. Dale, Jan 12 2015 *)

CROSSREFS

KEYWORD

nonn,easy

AUTHOR

Roman Witula, Aug 16 2012

EXTENSIONS

More terms from Harvey P. Dale, Jan 12 2015

STATUS

approved