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Search: id:A125585
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| A125585 |
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Array of constant-spaced integers read by antidiagonals. |
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+0
1
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| 1, 1, 2, 2, 3, 3, 1, 4, 5, 4, 2, 4, 6, 7, 5, 3, 5, 7, 8, 9, 6, 1, 6, 8, 10, 10, 11, 7, 2, 5, 9, 11, 13, 12, 13, 8, 3, 6, 9, 12, 14, 16, 14, 15, 9, 4, 7, 10, 13, 15, 17, 19, 16, 17, 10, 1, 8, 11, 14, 17, 18, 20, 22, 18, 19, 11, 2, 6, 12, 15, 18, 21, 21, 23, 25, 20, 21, 12
(list; graph; listen)
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OFFSET
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1,3
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COMMENT
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Iteratively taking sums of the values in each row starting with 1 produces the "figurate" numbers. For instance: 1, 1 + 2 = 3, 1 + 2 + 3 = 6 (the Triangle numbers -- A000217) 1, 1 + 3 = 4, 1 + 3 + 5 = 9 (the Square numbers -- A000290) 1, 1 + 4 = 5, 1 + 4 + 7 = 10 (the Pentagonal numbers -- A000326) etc. Iterative sums of the rows in between produce sequences related to the figurate numbers: 2, 2+4=6, 2+4+6=10 (oblong, or pronic, or heteromecic numbers -- A002378) 2, 2+5=7, 2+5+8=15 (second pentagonal numbers -- A005449) 3, 3+6=9, 3+6+9=18 (triangular matchstick numbers --A045943) etc. Iterative products produce the n-Factorial numbers: 1, 1*3=3, 1*3*5=15 (Double factorial numbers: (2n-1)!! -- A001147 2, 2*4=8, 2*4*6=48 (Double factorial numbers: (2n)!! -- A000165) 1, 1*4=4, 1*4*7=28, (Triple factorial numbers (3*n-2)!!! -- A007559) etc.
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EXAMPLE
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The array begins:
1..2..3..4..5..6
1..3..5..7..9..11
2..4..6..8..10.12
1..4..7..10.13.16
2..5..8..11.14.17
3..6..9..12.15.18
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CROSSREFS
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Cf. A000217, A000290, A000326, A002378, A005449, A045943, A001147, A000165, A007559.
Sequence in context: A071434 A128924 A116464 this_sequence A109973 A087175 A071820
Adjacent sequences: A125582 A125583 A125584 this_sequence A125586 A125587 A125588
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KEYWORD
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easy,nonn
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AUTHOR
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Andrew Plewe (aplewe(AT)sbcglobal.net), Jan 04 2007
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