login
The OEIS is supported by the many generous donors to the OEIS Foundation.

 

Logo
Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
A252864 Number of pairs in generation n of the tree T defined in Comments. 1
1, 1, 2, 3, 5, 8, 12, 18, 25, 35, 51, 75, 110, 161, 236, 346, 507, 743, 1089, 1596, 2339, 3428, 5024, 7363, 10791, 15815, 23178, 33969, 49784, 72962, 106931 (list; graph; refs; listen; history; text; internal format)
OFFSET
0,3
COMMENTS
Generation g(0) of T is (0,0). Thereafter, successive generations accrue according to the rule that if (j,k) is in T, then (j,k+1) and (k,j+k) are in T. An equivalent tree is generated as follows: start with the tree of polynomials, T*, having g(0) = 0 and rule that if p(x) is in T*, then p(x) + 1 and x*p(x) are in T*; then put x = (1+sqrt(5))/2, the golden ratio, and remove duplicates as they occur. Or, to obtain a third guise for T, in T* replace x^2 by x + 1 in every polynomial (e.g., replace x^3 by 2x+1, etc.), and remove duplicates as they occur.
Every ordered pair of nonnegative integers occurs exactly once in T.
LINKS
Christian Ballot, Clark Kimberling, and Peter J. C. Moses, Linear Recurrences Originating From Polynomial Trees, Fibonacci Quart. 55 (2017), no. 5, 15-27.
FORMULA
Conjecture: |g(n)| = |g(n-1)| + |g(n-3)| for n >= 12.
Empirical g.f.: (x-1)*(x^2+x+1)*(x^8+2*x^7+2*x^6+2*x^5+x^4+x^3+x^2+1) / (x^3+x-1). - Colin Barker, Feb 01 2015
EXAMPLE
Ordered pairs (i,j) are abbreviated as i,j in this list of 7 generations of T:
g(0): 0,0
g(1): 0,1
g(2): 0,2 1,1
g(3): 0,3 1,2 2,2
g(4): 0,4 1,3 2,3 2,4 3,3
g(5): 0,5 1,4 2,5 3,4 3,5 3,6 4,4 4,6
g(6): 0,6 1,5 2,6 3,7 4,5 4,7 4,8 5,5 5,7 5,8 6,9 6,10
MATHEMATICA
t = NestList[DeleteDuplicates[Flatten[Map[{# + {0, 1}, {Last[#], Total[#]}} &, #], 1]] &, {{0, 0}}, 30]; s[0] = t[[1]]; s[n_] := s[n] = Union[t[[n + 1]], s[n - 1]];
g[n_] := Complement[s[n], s[n - 1]]; g[0] = {{0, 0}};
Column[Table[g[z], {z, 0, 9}]]
Table[Length[g[z]], {z, 0, 10}]
CROSSREFS
Sequence in context: A127884 A105858 A299731 * A039899 A039901 A173564
KEYWORD
nonn,more
AUTHOR
Clark Kimberling, Jan 31 2015
STATUS
approved

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recents
The OEIS Community | Maintained by The OEIS Foundation Inc.

License Agreements, Terms of Use, Privacy Policy. .

Last modified April 16 22:53 EDT 2024. Contains 371755 sequences. (Running on oeis4.)