OFFSET
0,2
COMMENTS
LINKS
Clark Kimberling, Table of n, a(n) for n = 0..2000
Index entries for linear recurrences with constant coefficients, signature (1,0,1,1).
FORMULA
a(n) = a(n-1) + a(n-3) + a(n-4), where a(0) = 1, a(1) = 2, a(2) = 2, a(3) = 1.
G.f.: (-1 - x + 2*x^3)/(-1 + x + x^3 + x^4).
From Peter Bala, Nov 27 2021: (Start)
a(2*n) = 3*a(2*n-2) - a(2*n-4) - (-1)^n, for n >= 2;
a(2*n+1) = 3*a(2*n-1) - a(2*n-3) + 7*(-1)^n, for n >= 2.
a(4*n-1) = Fibonacci(2*n+1)^2 - Fibonacci(2*n)^2 + Fibonacci(2*n-1)^2 - Fibonacci(2*n-2)^2 - 3, for n >= 1;
a(4*n+1) = Fibonacci(2*n+2)^2 - Fibonacci(2*n+1)^2 + Fibonacci(2*n)^2 - Fibonacci(2*n-1)^2 + 3, for n >= 0.
Conjecture: a(2*n+7) = Fibonacci(n)^3*Sum_{k >= 1} k^2 * Fibonacci(n*k)/ Fibonacci(n+2)^(k+1), n >= 1. (End)
MATHEMATICA
LinearRecurrence[{1, 0, 1, 1}, {1, 2, 2, 1}, 100]
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Clark Kimberling, Nov 29 2017
STATUS
approved