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A160378
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a(n) = n^3 - n*(n+1)/2.
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9
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0, 0, 5, 21, 54, 110, 195, 315, 476, 684, 945, 1265, 1650, 2106, 2639, 3255, 3960, 4760, 5661, 6669, 7790, 9030, 10395, 11891, 13524, 15300, 17225, 19305, 21546, 23954, 26535, 29295, 32240, 35376, 38709, 42245, 45990, 49950, 54131, 58539
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OFFSET
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0,3
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COMMENTS
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The sum of the n-1 numbers between n^2 and n*(n+1) = a(n). - J. M. Bergot, Apr 15 2013
Use the terms in A061885 to form the antidiagonals for an array. The antidiagonals begin: 0;2,3;6,7,8;12,13,14,15;20,21,22,23,24,25. The sum of the terms in these antidiagonals = a(n)for n > 0. - J. M. Bergot, Jul 08 2013
a(n) is the sum of the n numbers strictly between n^2-n-1 and n^2. - Charlie Marion, Feb 21 2020
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LINKS
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Milan Janjic and B. Petkovic, A counting function, arXiv preprint arXiv:1301.4550 [math.CO], 2013.
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FORMULA
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a(n) = 4*a(n-1) - 6*a(n-2) + 4*a(n-3) - a(n-4) for n > 3.
G.f.: x^2*(5 + x)/(1 - x)^4. (End)
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EXAMPLE
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a(4) = 4^3 - 4*5/2 = 64 - 10 = 54.
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MATHEMATICA
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PROG
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(Magma) [ n^3-n*(n+1)/2: n in [0..50] ];
(SageMath) [n^3 -binomial(n+1, 2) for n in range(41)] # G. C. Greubel, Oct 14 2023
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CROSSREFS
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KEYWORD
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nonn,easy
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AUTHOR
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EXTENSIONS
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Definition clarified and offset changed from 1 to 0 by Klaus Brockhaus, Dec 12 2010
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STATUS
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approved
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