

A134509


Triangular numbers remaining after arranging positive terms of A000217 in a triangle and removing an infinite number of "^"shaped layers as described below.


2



15, 36, 45, 78, 105, 153, 210, 276, 325, 378, 465, 528, 561, 630, 741, 820, 861, 903, 990, 1128, 1225, 1275, 1326, 1378, 1485, 1653, 1770, 1830, 1953, 2016, 2145, 2346, 2485, 2556, 2775, 2850, 3003, 3240, 3403, 3486, 3655, 3828, 3916, 4095, 4371, 4560
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OFFSET

1,1


COMMENTS

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).


LINKS

Table of n, a(n) for n=1..46.


EXAMPLE

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 upsidedown "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, ....


CROSSREFS

Cf. A000217, A001844.
Sequence in context: A074891 A241282 A219096 * A062712 A224719 A033709
Adjacent sequences: A134506 A134507 A134508 * A134510 A134511 A134512


KEYWORD

nonn


AUTHOR

Rick L. Shepherd, Oct 28 2007


STATUS

approved



