OFFSET
1,1
COMMENTS
There are no such necklaces (or cycles).
Theorem (Berlekamp & Guy) There exists such a chain just if n = 9 or 11 or F_k or F_k - 1 for k > 3.
REFERENCES
B. Barwell, Problem 2732, Problems and conjectures, Journal of Recreational Mathematics 34 (2006), 220-223.
E. R. Berlekamp and R. K. Guy, Paper which MAY be called "Fibonacci plays Billiards" and which may be submitted to the Monthly (in preparation as of Jun 2011).
LINKS
Colin Barker, Table of n, a(n) for n = 1..1000
Andrii Arman, David S. Gunderson, Pak Ching Li, Properties of the Fibonacci-sum graph, arXiv:1710.10303 [math.CO], 2017.
Kyle Fox, William B. Kinnersley, Daniel McDonald, Nathan Orlow, Gregory J. Puleo, Spanning Paths in Fibonacci-sum Graphs, Fibonacci Quart. 52 (2014), 46-49.
Gregory Puleo, Fibsum Graphs, 2009.
Index entries for linear recurrences with constant coefficients, signature (0,2,0,0,0,-1).
FORMULA
G.f.: x*(3*x^13+2*x^12+3*x^11+3*x^10-4*x^9-2*x^8-2*x^7-3*x^6-2*x^5-x^4-x^3+3*x+2) / ((x-1)*(x+1)*(x^4+x^2-1)). - Colin Barker, Dec 02 2014
a(n) = 2*a(n-2) - a(n-6) for n>14. - Colin Barker, Dec 02 2014
EXAMPLE
Examples: 1 2; 1 2 3; 4 1 2 3; 4 1 2 3 5; 4 1 7 6 2 3 5; ...
MAPLE
S := {9, 11}: for i from 3 to 50 do S := S union {fibonacci(i)}: S := S union {fibonacci(i)-1}: od: S := S minus {1}: S := convert(S, list): S := sort(S):for i from 1 to nops(S) do printf(`%d, `, S[i]) od: # James A. Sellers, Feb 25 2003
PROG
(PARI) Vec(x*(3*x^13+2*x^12+3*x^11+3*x^10-4*x^9-2*x^8-2*x^7-3*x^6-2*x^5-x^4-x^3+3*x+2)/((x-1)*(x+1)*(x^4+x^2-1)) + O(x^100)) \\ Colin Barker, Dec 02 2014
(PARI) lista(nn) = Set(concat([9, 11], concat(vector(nn, n, fibonacci(n+3)), vector(nn, n, fibonacci(n+3)-1)))) \\ Michel Marcus, Oct 31 2017
CROSSREFS
KEYWORD
easy,nonn
AUTHOR
R. K. Guy, Feb 18 2003
EXTENSIONS
More terms from James A. Sellers, Feb 25 2003
Changes due to offset=1 from Colin Barker, Oct 31 2017
STATUS
approved