OFFSET
0,2
COMMENTS
Binomial transform of A001541.
Let A be the unit-primitive matrix (see [Jeffery]) A = A_(8,3) = [0,0,0,1; 0,0,2,0; 0,2,0,1; 2,0,2,0]. Then A084130(n) = (1/4)*Trace(A^(2*n)). (Cf. A006012, A001333.) - L. Edson Jeffery, Apr 04 2011
a(n) is also the rational part of the Q(sqrt*(2)) integer giving the length L(n) of a variant of the Lévy C-curve, given by _Kival Ngaokrajan_, at iteration step n. See A057084. - Wolfdieter Lang, Dec 18 2014
LINKS
G. C. Greubel, Table of n, a(n) for n = 0..1000
L. E. Jeffery, Unit-primitive matrices
Index entries for linear recurrences with constant coefficients, signature (8,-8).
FORMULA
a(n) = (4+sqrt(8))^n/2 + (4-sqrt(8))^n/2.
G.f.: (1-4*x)/(1-8*x+8*x^2).
E.g.f.: exp(4*x)*cosh(sqrt(8)*x).
From G. C. Greubel, Oct 13 2022: (Start)
a(2*n) = 2^(3*n-1)*A002203(2*n).
a(2*n+1) = 2^(3*n+2)*A000129(2*n+1). (End)
MATHEMATICA
LinearRecurrence[{8, -8}, {1, 4}, 30] (* Harvey P. Dale, Sep 25 2014 *)
PROG
(PARI) {a(n)= if(n<0, 0, real((4+ 2*quadgen(8))^n))}
(Magma) [n le 2 select 4^(n-1) else 8*(Self(n-1) -Self(n-2)): n in [1..41]]; // G. C. Greubel, Oct 13 2022
(SageMath)
A084130=BinaryRecurrenceSequence(8, -8, 1, 4)
[A084130(n) for n in range(41)] # G. C. Greubel, Oct 13 2022
CROSSREFS
KEYWORD
easy,nonn
AUTHOR
Paul Barry, May 16 2003
STATUS
approved