|
| |
|
|
A270572
|
|
a(1)=3; thereafter a(n) is the number of occurrences of a(n-1) in {a(1), ... , a(n-1)}.
|
|
1
|
|
|
|
3, 1, 1, 2, 1, 3, 2, 2, 3, 3, 4, 1, 4, 2, 4, 3, 5, 1, 5, 2, 5, 3, 6, 1, 6, 2, 6, 3, 7, 1, 7, 2, 7, 3, 8, 1, 8, 2, 8, 3, 9, 1, 9, 2, 9, 3, 10, 1, 10, 2, 10, 3, 11, 1, 11, 2, 11, 3, 12, 1, 12, 2, 12, 3, 13, 1, 13, 2, 13, 3, 14, 1, 14, 2, 14, 3, 15, 1, 15, 2, 15, 3, 16, 1, 16, 2, 16, 3, 17, 1, 17, 2, 17, 3, 18, 1, 18, 2, 18, 3, 19, 1
(list;
graph;
refs;
listen;
history;
text;
internal format)
|
|
|
|
OFFSET
|
1,1
|
|
|
COMMENTS
|
For a(1)=3 the members of the sequence from a(12) onwards are of the form a(6*r+6)=1, a(6*r+7)=3+r, a(6*r+8)=2, a(6*r+9)=3+r, a(6*r+10)=3, a(6*r+11)=4+r, r>=1.
For every integer a(1)>=1, from a(a(1)*a(1)+3) onwards, the members of the sequence are of the form a(n)=1, a(n+1)=a(1)+r, a(n+2)=2, a(n+3)=a(1)+r, ... , a(n+2*a(1)-2)=a(1), a(n+2*a(1)-1)=a(1)+r+1. n = a(1)*a(1)+2*a(1)*(r-1)+3, r>=1.
|
|
|
LINKS
|
|
|
|
FORMULA
|
a(n) = a(n-2)+a(n-6)-a(n-8) for n>9.
G.f.: x*(3+x-2*x^2+x^3+x^5-2*x^6-2*x^7+3*x^8+x^10-3*x^11-x^12+2*x^13-x^14) / ((1-x)^2*(1+x)^2*(1-x+x^2)*(1+x+x^2)).
(End)
|
|
|
EXAMPLE
|
a(1)=3. The number of occurrences of 3 in {3} is 1, thus a(2)=1. The number of occurrences of 1 in {3,1} is 1, thus a(3)=1. The number of occurrences of 1 in {3, 1, 1} is 2, thus a(4)=2. The number of occurrences of 2 in {3, 1, 1, 2} is 1, thus a(5)=1. The number of occurrences of 1 in {3, 1, 1, 2, 1} is 3, thus a(6)=3 , and so on.
|
|
|
MATHEMATICA
|
LinearRecurrence[{0, 1, 0, 0, 0, 1, 0, -1}, {3, 1, 1, 2, 1, 3, 2, 2, 3, 3, 4, 1, 4, 2, 4}, 120] (* Harvey P. Dale, Dec 16 2023 *)
|
|
|
PROG
|
(PARI) Vec(x*(3+x-2*x^2+x^3+x^5-2*x^6-2*x^7+3*x^8+x^10-3*x^11-x^12+2*x^13-x^14) / ((1-x)^2*(1+x)^2*(1-x+x^2)*(1+x+x^2)) + O(x^100)) \\ Colin Barker, Mar 20 2016
|
|
|
CROSSREFS
|
|
|
|
KEYWORD
|
nonn,easy
|
|
|
AUTHOR
|
|
|
|
STATUS
|
approved
|
| |
|
|
