OFFSET
0,3
COMMENTS
Apparently (for n > 0), numbers that have a unique partition into a sum of distinct Lucas numbers (A000204).
LINKS
Eric Weisstein's World of Mathematics, Lucas Number
Index entries for linear recurrences with constant coefficients, signature (1,2,-1,-1).
FORMULA
G.f.: x*(1 + 2*x)/((1 - x)*(1 + x)*(1 - x - x^2)).
a(n) = a(n-1) + 2*a(n-2) - a(n-3) - a(n-4).
a(n) = Lucas(n+1) - (3 - (-1)^n)/2.
a(n) = floor(phi^(n+1)) - 1, where phi = (1 + sqrt(5))/2 is the golden ratio (A001622).
a(n) = Sum_{k>=0} A051601(n-k,k) (conjectured). - Greg Dresden, May 18 2023
MATHEMATICA
CoefficientList[Series[x (1 + 2 x)/((1 - x) (1 + x) (1 - x - x^2)) , {x, 0, 40}], x]
LinearRecurrence[{1, 2, -1, -1}, {0, 1, 3, 5}, 41]
Table[LucasL[n + 1] - (3 - (-1)^n)/2, {n, 0, 40}]
Table[Floor[GoldenRatio^(n + 1)] - 1, {n, 0, 40}]
PROG
(PARI) a(n) = fibonacci(n) + fibonacci(n+2) + ((-1)^n - 3)/2; \\ Altug Alkan, Mar 25 2018
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Ilya Gutkovskiy, Mar 25 2018
STATUS
approved