1,1

Let "s" be any sequence (finite or infinite) and "b" be any set of real numbers. Define an operation 'triangular removal', TriRem(s,b), that produces a subsequence from s as follows: Arrange the terms s(i) by rows into a triangle, which can be viewed as a (possibly infinite) set of nested "^"-shaped layers. Count each layer from the outside as layer 1, 2, 3, .... During the following removal process, these layer numbers are considered fixed: For each positive integer n in b, remove layer n if it exists. TriRem(s,b) is the sequence of remaining terms read by rows. The current sequence, A134509, is TriRem(A000217-{0},A000217). A complementary operation 'triangular retention', TriRet(s,b), can be defined similarly that instead retains the layers specified by b. The index of an original term s(i) at the apex of a removed/retained "^"-layer is a centered square number (A001844).

The original triangle of positive triangular numbers begins like this:

........................1

......................3...6

....................10..15..21

..................28..36..45..55

................66..78..91..105.120

..............136.153.171.190.210.231

......................................

The upside-down "V" with 1 at the top is layer 1, with 15 at the top is layer 2, with 91 at the top is layer 3, etc. Because 1 and 3 are elements of b=A000217, layers 1 and 3 are among those completely removed. The remaining terms by row begin the infinite subsequence: 15, 36, 45, 78, 105, ....

Cf. A000217, A001844.

Adjacent sequences: A134506 A134507 A134508 this_sequence A134510 A134511 A134512

Sequence in context: A107423 A027442 A074891 this_sequence A062712 A033709 A067707

nonn

Rick L. Shepherd (rshepherd2(AT)hotmail.com), Oct 28 2007