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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

Table of n, a(n) for n=0..30.

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: A111388 A127884 A105858 * A039899 A039901 A173564

Adjacent sequences:  A252861 A252862 A252863 * A252865 A252866 A252867

KEYWORD

nonn,easy,more

AUTHOR

Clark Kimberling, Jan 31 2015

STATUS

approved

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Last modified June 28 09:49 EDT 2017. Contains 288813 sequences.