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A253145
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Triangular numbers (A000217) omitting the term 1.
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3
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0, 3, 6, 10, 15, 21, 28, 36, 45, 55, 66, 78, 91, 105, 120, 136, 153, 171, 190, 210, 231, 253, 276, 300, 325, 351, 378, 406, 435, 465, 496, 528, 561, 595, 630, 666, 703, 741, 780, 820, 861, 903, 946, 990, 1035, 1081, 1128, 1176, 1225, 1275
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OFFSET
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0,2
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COMMENTS
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The full triangle of the inverse Akiyama-Tanigawa transform applied to (-1)^n*A062510(n)=3*(-1)^n*A001045(n) yielding a(n) is
0, 3, 6, 10, 15, 21, 28, 36, ...
-3, -6, -12, -20, -30, -42, -56, ... essentially -A002378
3, 12, 24, 40, 60, 84, ... essentially A046092
-9, -24, -48, -80, -120, ... essentially -A033996
15, 48, 96, 160, ...
-33, -96, -192, ...
63, 192, ...
-129, ...
etc.
An autosequence of the first kind is a sequence whose main diagonal is A000004 = 0's.
b(n) = 0, 0 followed by a(n) is an autosequence of the first kind.
The successive differences of b(n) are
0, 0, 0, 3, 6, 10, 15, 21, ...
0, 0, 3, 3, 4, 5, 6, 7, ... see A194880(n)
0, 3, 0, 1, 1, 1, 1, 1, ...
3, -3, 1, 0, 0, 0, 0, 0, ...
-6, 4, -1, 0, 0, 0, 0, 0, ...
10, -5, 1, 0, 0, 0, 0, 0, ...
-15, 6, -1, 0, 0, 0, 0, 0, ...
21, -7, 1, 0, 0, 0, 0, 0, ...
The inverse binomial transform (first column) is the signed sequence. This is general.
Also generalized hexagonal numbers without 1. - Omar E. Pol, Mar 23 2015
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LINKS
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FORMULA
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Inverse Akiyama-Tanigawa transform of (-1)^n*A062510(n).
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MATHEMATICA
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PROG
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(GAP) Concatenation([0], List([1..50], n->(n+1)*(n+2)/2)); # Muniru A Asiru, Oct 31 2018
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CROSSREFS
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Cf. A000217, A179865, A001045, A062510, A002378, A046092, A033996, A122803, A007283, A091629, A020714, A110286, A161680, A175805, A005010, A194880, A001840, A022003, A255935.
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KEYWORD
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nonn,easy
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AUTHOR
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STATUS
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approved
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