A274168 Number of real integers in n-th generation of tree T(r) defined in Comments, where r^2 = -r - 1 (i.e., r = (-1 + sqrt(3))/2).

1, 1, 1, 1, 1, 1, 2, 2, 3, 3, 5, 5, 7, 9, 13, 18, 25, 33, 43

Offset

0,7

Comments

Let T* be the infinite tree with root 0 generated by these rules: if p is in T*, then p+1 is in T* and x*p is in T*. Let g(n) be the set of nodes in the n-th generation, so that g(0) = {0}, g(1) = {1}, g(2) = {2,x}, g(3) = {3,2x,x+1,x^2}, etc. Let T(r) be the tree obtained by substituting r for x.

See A274142 for a guide to related sequences.

Links

Table of n, a(n) for n=0..18.

Example

If r = (-1 + sqrt(3))/2, then g(3) = {3,2r,r+1,r^2}, in which the number of real integers is a(3) = 1.

Mathematica

z = 18; t = Join[{{0}}, Expand[NestList[DeleteDuplicates[Flatten[Map[{# + 1, x*#} &, #], 1]] &, {1}, z]]];
u = Table[t[[k]] /. x -> (-1 + 3 I)/2, {k, 1, z}]; Table[
Count[Map[IntegerQ, u[[k]]], True], {k, 1, z}]

Crossrefs

Cf. A274142.

Sequence in context: A140829 A346523 A184324 * A116575 A244800 A275972

Adjacent sequences: A274165 A274166 A274167 * A274169 A274170 A274171

Keyword

nonn,more

Author

Clark Kimberling, Jun 13 2016

Status

approved