A338994 Table read by antidiagonals: if x(n+1) = A001414(x(n-1)) + 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.

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

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

Robert Israel, Table of n, a(n) for n = 1..10011 (first 141 antidiagonals, flattened)

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, k-i]), i=1..k-1), k=2..14);

Crossrefs

Cf. A001414, A338937.

Sequence in context: A273466 A278938 A303157 * A217394 A072998 A024629

Adjacent sequences: A338991 A338992 A338993 * A338995 A338996 A338997

Keyword

nonn,tabl

Author

J. M. Bergot and Robert Israel, Nov 17 2020

Status

approved