A274185 -id:A274185 - cp's OEIS frontend

A274183 Irregular triangular array having n-th row g(n) defined in Comments.

0, 1, 0, 2, 1, 0, 3, 2, 1, 1, 0, 4, 3, 2, 2, 1, 1, 0, 5, 4, 3, 3, 2, 2, 1, 2, 1, 1, 0, 6, 5, 4, 4, 3, 3, 2, 3, 2, 2, 1, 2, 1, 1, 1, 0, 7, 6, 5, 5, 4, 4, 3, 4, 3, 3, 2, 3, 2, 2, 2, 1, 3, 2, 2, 1, 1, 1, 1, 0, 8, 7, 6, 6, 5, 5, 4, 5, 4, 4, 3, 4, 3, 3, 3, 2, 4

Offset: 0

Clark Kimberling, Jun 13 2016

Let g(0) = (0) and for n > 0, define g(n) inductively as the concatenation of g(n-1) and the numbers k/2 as k ranges through the even numbers k in g(n-1). Every nonnegative integer appears infinitely many times. For the limiting ratio of lengths of consecutive rows, see A274192.

			First six rows:
0
1   0
2   1   0
3   2   1   1   0
4   3   2   2   1   1   0
5   4   3   3   2   2   1   2   1   1   0

Cf. A274184 (row lengths), A274192, A274185.

g[0] = {0}; z = 14; g[n_] := g[n] = Join[g[n - 1] + 1, (1/2) Select[g[n - 1], IntegerQ[#/2] &]]; Flatten[Table[g[n], {n, 0, z}]]

A274149 Number of integers in n-th generation of tree T(-1/4) defined in Comments.

Original entry on oeis.org

1, 1, 1, 1, 1, 2, 2, 3, 4, 6, 7, 9, 12, 17, 22, 29, 38, 51, 68, 90, 119, 158, 209, 277, 368, 489, 648, 858, 1137, 1509, 2002, 2655, 3520, 4667, 6189, 8208, 10885, 14436, 19141, 25382, 33659, 44638, 59195, 78497, 104092, 138036, 183050, 242745, 321904, 426875

Offset: 0

Clark Kimberling, Jun 11 2016

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.

			For r = -1/4, we have g(3) = {3,2r,r+1, r^2}, in which the number of integers is a(3) = 1.

Kenny Lau, Table of n, a(n) for n = 0..8153

Cf. A274142.

z = 18; t = Join[{{0}}, Expand[NestList[DeleteDuplicates[Flatten[Map[{# + 1, x*#} &, #], 1]] &, {1}, z]]];
u = Table[t[[k]] /. x -> -1/4, {k, 1, z}];
Table[Count[Map[IntegerQ, u[[k]]], True], {k, 1, z}]

a(n-1) = length of row n of the array in A274185.

More terms from Kenny Lau, Jul 01 2016

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A274183 Irregular triangular array having n-th row g(n) defined in Comments.

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