OFFSET
1,4
COMMENTS
Any sequence a(1),a(2),a(3),... defined by the recurrence a(n) = (a(n-1) + a(n-2) + 1)/a(n-3) (for n>3) has period 8. - James Propp, Nov 20 2002. This is the 8-cycle discovered by H. Todd - see Lyness, Note 1847. - N. J. A. Sloane, Jul 19 2020
LINKS
R. C. Lyness, Note 1581. Cycles, Math. Gazette, 26 (1942), 62.
R. C. Lyness, Note 1847. Cycles, Math. Gaz., 29 (1945), 231-233.
R. C. Lyness, Note 2952. Cycles, Math. Gaz., 45 (1961), 207-209.
Index entries for linear recurrences with constant coefficients, signature (0, 0, 0, 0, 0, 0, 0, 1).
MAPLE
a := 1; b := 1; c := 1; f := proc(n) option remember; global a, b, c; if n=1 then RETURN(a); fi; if n=2 then RETURN(b); fi; if n=3 then RETURN(c); fi; RETURN((f(n-1)+f(n-2)+1)/f(n-3)); end;
MATHEMATICA
nxt[{a_, b_, c_}]:={b, c, (b+c+1)/a}; Transpose[NestList[nxt, {1, 1, 1}, 110]][[1]] (* or *) PadRight[{}, 110, {1, 1, 1, 3, 5, 9, 5, 3}] (* Harvey P. Dale, Jan 13 2015 *)
LinearRecurrence[{0, 0, 0, 0, 0, 0, 0, 1}, {1, 1, 1, 3, 5, 9, 5, 3}, 105] (* Ray Chandler, Aug 25 2015 *)
CROSSREFS
KEYWORD
nonn
AUTHOR
N. J. A. Sloane, Nov 21 2002
STATUS
approved