|
|
A295690
|
|
a(n) = a(n-1) + a(n-3) + a(n-4), where a(0) = 2, a(1) = 2, a(2) = 1, a(3) = 1.
|
|
2
|
|
|
2, 2, 1, 1, 5, 8, 10, 16, 29, 47, 73, 118, 194, 314, 505, 817, 1325, 2144, 3466, 5608, 9077, 14687, 23761, 38446, 62210, 100658, 162865, 263521, 426389, 689912, 1116298, 1806208, 2922509, 4728719, 7651225, 12379942, 20031170, 32411114, 52442281, 84853393
(list;
graph;
refs;
listen;
history;
text;
internal format)
|
|
|
OFFSET
|
0,1
|
|
COMMENTS
|
a(n)/a(n-1) -> (1 + sqrt(5))/2 = golden ratio (A001622), so that a( ) has the growth rate of the Fibonacci numbers (A000045).
|
|
LINKS
|
|
|
FORMULA
|
a(n) = a(n-1) + a(n-3) + a(n-4), where a(0) = 2, a(1) = 2, a(2) = 1, a(3) = 1.
G.f.: (-2 + x^2 + 2 x^3)/(-1 + x + x^3 + x^4).
a(2*n) = (3/5)*Lucas(2*n) + (4/5)*(-1)^n.
a(2*n+1) = (3/5)*Lucas(2*n+1) + (7/5)*(-1)^n.
a(2*n) = a(2*n-1) + a(2*n-2) + 3*(-1)^n.
a(2*n+1) = a(2*n) + a(2*n-1) + 2*(-1)^n.
a(2*n+1)*F(n+3) - a(2*n+3)*F(n-1) = 3*F(n+1)^3, where F(n) = A000045(n). (End)
|
|
MATHEMATICA
|
LinearRecurrence[{1, 0, 1, 1}, {2, 2, 1, 1}, 100]
|
|
PROG
|
(Magma) a:=[2, 2, 1, 1]; [n le 4 select a[n] else Self(n-1) + Self(n-3) + Self(n-4):n in [1..40]]; // Marius A. Burtea, Nov 13 2019
|
|
CROSSREFS
|
|
|
KEYWORD
|
nonn,easy
|
|
AUTHOR
|
|
|
STATUS
|
approved
|
|
|
|