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A214683 a(n+3) = -a(n+2) + 2a(n+1) + a(n) with a(0)=-1, a(1)=0, a(2)=-3. 18
-1, 0, -3, 2, -8, 9, -23, 33, -70, 113, -220, 376, -703, 1235, -2265, 4032, -7327, 13126, -23748, 42673, -77043, 138641, -250054, 450293, -811760, 1462292, -2635519, 4748343, -8557089, 15418256, -27784091, 50063514, -90213440, 162556377, -292919743 (list; graph; refs; listen; history; text; internal format)
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

Table of n, a(n) for n=0..34.

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

CROSSREFS

Cf. A006053.

Sequence in context: A021308 A274181 A195055 * A060921 A163356 A209360

Adjacent sequences:  A214680 A214681 A214682 * A214684 A214685 A214686

KEYWORD

sign,easy

AUTHOR

Roman Witula, Jul 25 2012

STATUS

approved

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Last modified November 15 18:59 EST 2019. Contains 329149 sequences. (Running on oeis4.)