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 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 (list; graph; refs; listen; history; text; internal format)
 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 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 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, .... CROSSREFS Cf. A000217, A001844. Sequence in context: A241282 A249056 A219096 * A062712 A224719 A033709 Adjacent sequences:  A134506 A134507 A134508 * A134510 A134511 A134512 KEYWORD nonn AUTHOR Rick L. Shepherd, Oct 28 2007 STATUS approved

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