OFFSET
0,3
COMMENTS
Ramanujan-type sequence number 1 for the argument 2Pi/7.
The discussed sequence is associated with the sequence A006053 (with respect to the similar trigonometric formulas describing both sequences). Indeed, we have 7^(1/3)*a(n) = (c(1)/c(2))^(1/3)*(2c(1))^n + (c(2)/c(4))^(1/3)*(2c(2))^n + (c(4)/c(1))^(1/3)*(2c(4))^n = (c(1)/c(2))^(1/3)*(2c(2))^(n+1) + (c(2)/c(4))^(1/3)*(2c(4))^(n+1) + (c(4)/c(1))^(1/3)*(2c(1))^(n+1), where c(j) := Cos(2Pi*j/7), which is "almost" the copy of the respective formula for A006053. From a(0), A006053(0) and a(1), A006053(1), (and again) A006053(0) we deduce the following attractive decompositions
x^3 - 7^(1/3)*x - 1 = (x - (c(1)/c(4))^(1/3))*(x - (c(2)/c(1))^(1/3))*(x - (c(4)/c(2))^(1/3)), and
x^3 - 49^(1/3)*x - 1 = (x - (c(1)/c(2))^(1/3)*2c(1))*(x - (c(2)/c(4))^(1/3)*2c(2))*(x - (c(4)/c(1))^(1/3)*2c(4)).
From Newton-Girard formulas applied to these polynomials we generate two new sequences of real numbers S(n) := (c(1)/c(4))^(n/3) + (c(2)/c(1))^(n/3) + (c(4)/c(2))^(n/3), and T(n) := ((c(1)/c(2))^(1/3)*2c(1))^n + ((c(2)/c(4))^(1/3)*2c(2))^n + ((c(4)/c(1))^(1/3)*2c(4))^n. In first Witula's paper it is proved that S(n) = as(n) + bs(n)*7^(1/3) + cs(n)*49^(1/3), where as(n+3) = as(n) + 7cs(n+1), bs(n+3) = bs(n) + as(n+1), cs(n+3) = cs(n) + bs(n+1), as(0)=3, as(1)=as(2)=bs(0)=bs(1)=0, bs(2)=2, cs(0)=cs(1)=cs(2)=0, and T(n) = at(n) + bt(n)*7^(1/3) + ct(n)*49^(1/3), where at(n+3) = at(n) + 7bt(n+1), bt(n+3) = bt(n) + 7ct(n+1), ct(n+3) = ct(n) + at(n+1), at(0)=3, at(1)=at(2)=bt(0)=bt(1)=bt(2)=ct(0)=ct(1)=0, ct(2)=2. All six sequences as(n),bs(n),...,ct(n) are created from integers and will be discussed in separate sequences .
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
G. C. Greubel, Table of n, a(n) for n = 0..1000
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 (-1,2,1).
FORMULA
a(n+3) + a(n+2) - 2a(n+1) - a(n) = 0, a(0)=-1, a(1)=0, a(2)=-3.
G.f.: -(1+x+x^2)/(1+x-2*x^2-x^3).
EXAMPLE
From values of a(k), for k=0,1,..,5 we deduce that (c(1)/c(2))^(1/3)*A + (c(2)/c(4))^(1/3)*B + (c(4)/c(1))^(1/3)*C = 0 in the following cases: A=2c(1), B=2c(2), C=2c(4) or A=-1+(2c(1))^2+(2c(1))^3, B=-1+(2c(2))^2+(2c(2))^3, C=-1+(2c(3))^2+(2c(3))^3 or A=1+(2c(1))^4+(2c(1))^5, B=1+(2c(2))^4+(2c(2))^5, C=1+(2c(3))^4+(2c(3))^5.
MATHEMATICA
LinearRecurrence[{-1, 2, 1}, {-1, 0, -3}, 40]
PROG
(Magma) a:=[-1, 0, -3]; [ n le 3 select a[n] else -Self(n-1) + 2*Self(n-2) + Self(n-3): n in [1..35]]; // Marius A. Burtea, Oct 03 2019
(SageMath)
@CachedFunction
def a(n): # a = A214683
if (n<3): return (-1, 0, -3)[n]
else: return -a(n-1) + 2*a(n-2) + a(n-3)
[a(n) for n in range(40)] # G. C. Greubel, Nov 25 2022
CROSSREFS
KEYWORD
sign,easy
AUTHOR
Roman Witula, Jul 25 2012
STATUS
approved