OFFSET
0,2
COMMENTS
Satisfies of recurrence relations system a(n) = 3*a(n-1) + 2*b(n-1), b(n) = b(n-1) - 2*a(n-1), a(0)=1, b(0)=2.
More generally, for the recurrence relations system a(n) = 3*a(n-1) + 2*b(n-1), b(n) = b(n-1) - 2*a(n-1), a(0)=k, b(0)=m solution is a(n) = ((2 + i*sqrt(3))^n*((sqrt(3) - i)*k - 2*i*m) + (2 - i*sqrt(3))^n*((sqrt(3) + i)*k + 2*i*m))/(2*sqrt(3)), b(n) = ((2 - i*sqrt(3))^n*((sqrt(3) - i)*m - 2*i*k) + (2 + i*sqrt(3))^n*(2*i*k + (sqrt(3) + i)*m))/(2*sqrt(3)), where i is the imaginary unit.
LINKS
FORMULA
O.g.f.: (1 + 3*x)/(1 - 4*x + 7*x^2).
E.g.f.: (5*sqrt(3)*sin(sqrt(3)*x) + 3*cos(sqrt(3)*x))*exp(2*x)/3.
a(n) = 4*a(n-1) - 7*a(n-2).
a(n) = ((2 + i*sqrt(3))^n*(-5*i + sqrt(3)) + (2 - i*sqrt(3))^n*(5*i + sqrt(3)))/(2*sqrt(3)), where i is the imaginary unit.
a(n) = 7^(n/2)*((5/sqrt(3))*sin(c)+cos(c)) with c = n*arctan(sqrt(3)/2). - Peter Luschny, Jul 21 2016
MAPLE
a:=series((1+3*x)/(1-4*x+7*x^2), x=0, 30): seq(coeff(a, x, n), n=0..29); # Paolo P. Lava, Mar 27 2019
MATHEMATICA
LinearRecurrence[{4, -7}, {1, 7}, 30]
PROG
(PARI) Vec((1+3*x)/(1-4*x+7*x^2) + O(x^99)) \\ Altug Alkan, Jul 13 2016
CROSSREFS
KEYWORD
sign,easy
AUTHOR
Ilya Gutkovskiy, Jul 13 2016
STATUS
approved