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Search: id:A130684
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| A130684 |
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Triangular array T read by rows, where T(n, k) is the number of squares (not necessarily orthogonal) all of whose vertices lie in a (n + 1) by (k + 1) square lattice. |
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1
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| 1, 2, 6, 3, 10, 20, 4, 14, 30, 50, 5, 18, 40, 70, 105, 6, 22, 50, 90, 140, 196, 7, 26, 60, 110, 175, 252, 336, 8, 30, 70, 130, 210, 308, 420, 540, 9, 34, 80, 150, 245, 364, 504, 660, 825, 10, 38, 90, 170, 280, 420, 588, 780, 990, 1210, 11, 42, 100, 190, 315, 476, 672
(list; table; graph; listen)
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OFFSET
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1,2
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COMMENT
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Rows are 1; 2, 6; 3, 10, 20; 4, 14, 30, 50; ... Reading down the long diagonal gives A002415.
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LINKS
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Joel Lewis (jblewis(AT)post.harvard.edu), Jun 29 2007, Table of n, a(n) for n = 1..210
Problem solved on the Art of Problem Solving forum, Number of squares in a grid
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FORMULA
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n(n+1)(n+2)(2n - k + 1)/12 (k <= n)
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EXAMPLE
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T(2, 2) = 6 because there are 6 squares all of whose vertices lie in a 3 X 3 lattice: four squares of side length 1, one square of side length 2 and one non-orthogonal square of side length the square root of 2.
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CROSSREFS
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Cf. A002415.
Sequence in context: A079297 A109465 A090705 this_sequence A079178 A113552 A064736
Adjacent sequences: A130681 A130682 A130683 this_sequence A130685 A130686 A130687
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KEYWORD
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easy,nonn,tabl
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AUTHOR
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Joel Lewis (jblewis(AT)post.harvard.edu), Jun 29 2007
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