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A083374
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a(n) = n^2 * (n^2 - 1)/2.
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38
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0, 6, 36, 120, 300, 630, 1176, 2016, 3240, 4950, 7260, 10296, 14196, 19110, 25200, 32640, 41616, 52326, 64980, 79800, 97020, 116886, 139656, 165600, 195000, 228150, 265356, 306936, 353220, 404550, 461280, 523776, 592416, 667590, 749700, 839160, 936396
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OFFSET
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1,2
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COMMENTS
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Triangular numbers t_n as n runs through the squares.
Partial sums of A055112: If one generated Pythagorean primitive triangles from n, n+1, then the collective areas of n of them would be equal to the numbers in this sequence. The sum of the first three triangles is 6+30+84 = 120 which is the third nonzero term of the sequence. - J. M. Bergot, Jul 14 2011
a(n) is the number of segments on an n X n grid or geoboard. - Martin Renner, Apr 17 2014
Consider the partitions of 2n into two parts (p,q). Then a(n) is the total volume of the family of rectangular prisms with dimensions p, q and |q-p|. - Wesley Ivan Hurt, Apr 15 2018
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REFERENCES
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Albert H. Beiler, Recreations in the theory of numbers, New York: Dover, (2nd ed.) 1966, p. 106, table 55.
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LINKS
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FORMULA
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a(n) = 5*a(n-1)-10*a(n-2)+10*a(n-3)-5*a(n-4)+a(n-5) for n>5. - R. J. Mathar, Apr 10 2009
Sum_{n>=2} (-1)^n/a(n) = Pi^2/6 - 3/2. - Amiram Eldar, Nov 02 2021
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MAPLE
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MATHEMATICA
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Table[n^2*(n^2-1)/2, {n, 40}] (* T. D. Noe, Oct 25 2006 *)
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PROG
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CROSSREFS
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KEYWORD
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easy,nonn
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AUTHOR
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Alan Sutcliffe (alansut(AT)ntlworld.com), Jun 05 2003
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EXTENSIONS
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Corrected and extended by T. D. Noe, Oct 25 2006
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STATUS
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approved
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