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Number of pairs in generation n of the tree T defined in Comments.
1

%I #28 May 19 2020 02:28:34

%S 1,1,2,3,5,8,12,18,25,35,51,75,110,161,236,346,507,743,1089,1596,2339,

%T 3428,5024,7363,10791,15815,23178,33969,49784,72962,106931

%N Number of pairs in generation n of the tree T defined in Comments.

%C 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.

%C Every ordered pair of nonnegative integers occurs exactly once in T.

%H Christian Ballot, Clark Kimberling, and Peter J. C. Moses, <a href="https://www.fq.math.ca/Papers1/55-5/BallotKimberlingMoses.pdf">Linear Recurrences Originating From Polynomial Trees</a>, Fibonacci Quart. 55 (2017), no. 5, 15-27.

%F Conjecture: |g(n)| = |g(n-1)| + |g(n-3)| for n >= 12.

%F 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

%e Ordered pairs (i,j) are abbreviated as i,j in this list of 7 generations of T:

%e g(0): 0,0

%e g(1): 0,1

%e g(2): 0,2 1,1

%e g(3): 0,3 1,2 2,2

%e g(4): 0,4 1,3 2,3 2,4 3,3

%e g(5): 0,5 1,4 2,5 3,4 3,5 3,6 4,4 4,6

%e 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

%t 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]];

%t g[n_] := Complement[s[n], s[n - 1]]; g[0] = {{0, 0}};

%t Column[Table[g[z], {z, 0, 9}]]

%t Table[Length[g[z]], {z, 0, 10}]

%K nonn,more

%O 0,3

%A _Clark Kimberling_, Jan 31 2015