OFFSET
0,1
COMMENTS
The tetranacci sequence(s) (T(n+3) = T(n+2) + T(n+1) + T(n)) have characteristic roots such that one is real and the other two have imaginary components. The real root is given by r = (1/3)*(1 + x + y), where x = (19 + 3*sqrt(33))^(1/3) and y = (19 - 3*sqrt(33))^(1/3). By increasing the power of this root (r^n) it is found that r^n = (1/3)*( A001644(n) + (1/3)*a(n-1)*(x + y) + (1/3)*A000073(n-2)*(x^2 + y^2)). Similar results can be found for the complex roots.
LINKS
G. C. Greubel, Table of n, a(n) for n = 0..1000
Pin-Yen Lin, De Moivre type identities for the Tribonacci numbers, The Fibonacci Quarterly 26, no.2, (1988), 131-134
Index entries for linear recurrences with constant coefficients, signature (1,1,1).
FORMULA
EXAMPLE
a(4) = a(3) + a(2) + a(1) = 10 + 5 + 2 = 17. - Michael B. Porter, Jul 05 2016
MATHEMATICA
CoefficientList[Series[(3-x)/(1-x-x^2-x^3), {x, 0, 40}], x] (* or *) LinearRecurrence[{1, 1, 1}, {3, 2, 5}, 40]
PROG
(PARI) Vec((3-x)/(1-x-x^2-x^3) + O(x^40)) \\ Altug Alkan, Jul 04 2016
(Magma) R<x>:=PowerSeriesRing(Integers(), 40); Coefficients(R!( (3-x)/(1-x-x^2-x^3) )); // G. C. Greubel, Apr 23 2019
(Sage) ((3-x)/(1-x-x^2-x^3)).series(x, 40).coefficients(x, sparse=False) # G. C. Greubel, Apr 23 2019
(GAP) a:=[3, 2, 5];; for n in [4..40] do a[n]:=a[n-1]+a[n-2]+a[n-3]; od; a; # G. C. Greubel, Apr 23 2019
CROSSREFS
KEYWORD
nonn
AUTHOR
G. C. Greubel, Jul 04 2016
STATUS
approved