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 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 = t[]; s[n_] := s[n] = Union[t[[n + 1]], s[n - 1]]; g[n_] := Complement[s[n], s[n - 1]]; g = {{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 Adjacent sequences:  A252861 A252862 A252863 * A252865 A252866 A252867 KEYWORD nonn,more,changed AUTHOR Clark Kimberling, Jan 31 2015 STATUS approved

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Last modified May 26 10:08 EDT 2020. Contains 334620 sequences. (Running on oeis4.)