

A338994


Table read by antidiagonals: if x(n+1) = A001414(x(n1)) + A001414(x(n)) with x(0) = i and x(1) = j, then T(i,j) is the first k such that (x(k), x(k+1)) is a fixed point or a member of a cycle. If there is no such k, then T(i,j) = 1.


1



2, 20, 21, 19, 19, 20, 15, 18, 10, 16, 10, 18, 18, 10, 11, 10, 9, 14, 9, 16, 11, 8, 9, 17, 14, 15, 16, 9, 14, 15, 17, 17, 14, 15, 15, 11, 14, 14, 8, 17, 9, 14, 9, 15, 11, 8, 14, 14, 13, 9, 9, 13, 15, 15, 9, 13, 15, 14, 13, 8, 9, 9, 14, 15, 15, 14, 8, 12, 8, 13, 16, 8, 9, 13, 14, 9, 7, 9, 9, 15, 8
(list;
table;
graph;
refs;
listen;
history;
text;
internal format)



OFFSET

1,1


COMMENTS

The fixed points are (0,0) and (16,16) (i.e., if x(0)=16 and x(1)=16 then all x(n)=16). Cycles include (23, 32, 33, 24), (19, 28, 30, 21, 20), and (23, 34, 42, 31, 43, 74, 82, 82, 86, 88, 62, 50, 45).
Are there other cycles? Is T(i,j) ever 1? For 1 <= i <= 3000 and 1 <= j <= 3000, T(i,j) is never 1 and no other cycles are encountered.


LINKS



EXAMPLE

Table begins
2, 20, 19, 15, 10, 10, 8, 14, 14, 8, 13, 8, ...
21, 19, 18, 18, 9, 9, 15, 14, 14, 15, 12, 15, ...
20, 10, 18, 14, 17, 17, 8, 14, 14, 8, 8, 8, ...
16, 10, 9, 14, 17, 17, 13, 13, 13, 13, 12, 13, ...
11, 16, 15, 14, 9, 9, 8, 16, 16, 8, 12, 8, ...
11, 16, 15, 14, 9, 9, 8, 16, 16, 8, 12, 8, ...
9, 15, 9, 13, 9, 9, 7, 14, 14, 7, 12, 7, ...
11, 15, 15, 14, 13, 13, 12, 13, 13, 12, 15, 12, ...
11, 15, 15, 14, 13, 13, 12, 13, 13, 12, 15, 12, ...
9, 15, 9, 13, 9, 9, 7, 14, 14, 7, 12, 7, ...
14, 7, 6, 6, 23, 23, 4, 16, 16, 4, 12, 4, ...
9, 15, 9, 13, 9, 9, 7, 14, 14, 7, 12, 7, ...
T(1,7) = 8 because starting at x(0)=1, x(1)=7 we have x(2)=7, x(3)=14, x(4)=16, x(5)=17, x(6)=25, x(7)=27, x(8)=19, x(9)=28, and (19,28) is in the cycle (19, 28, 30, 21, 20).


MAPLE

spf:= proc(n) local t; add(t[1]*t[2], t=ifactors(n)[2]) end proc:
Cyc:= {[0, 0], [16, 16], [32, 33], [33, 24], [24, 23], [23, 32], [28, 30], [30, 21], [21, 20], [20, 19], [19, 28], [34, 42], [42, 31], [31, 43], [43, 74], [74, 82], [82, 82], [82, 86], [86, 88], [88, 62], [62, 50], [50, 45], [45, 23], [23, 34]}:
f:= proc(t) local count, x;
count:= 0;
x:= t;
while count < 1000 do
if member(x, Cyc) then return count fi;
x:= [x[2], spf(x[1])+spf(x[2])];
count:= count+1;
od;
FAIL
end proc:
seq(seq(f([i, ki]), i=1..k1), k=2..14);


CROSSREFS



KEYWORD



AUTHOR



STATUS

approved



