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A087059
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Difference between 2*n^2 and the next greater square number.
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10
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2, 1, 7, 4, 14, 9, 2, 16, 7, 25, 14, 1, 23, 8, 34, 17, 47, 28, 7, 41, 18, 56, 31, 4, 46, 17, 63, 32, 82, 49, 14, 68, 31, 89, 50, 9, 71, 28, 94, 49, 2, 72, 23, 97, 46, 124, 71, 16, 98, 41, 127, 68, 7, 97, 34, 128, 63, 161, 94, 25, 127, 56, 162, 89, 14, 124, 47, 161, 82, 1, 119
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OFFSET
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1,1
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COMMENTS
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For n >= 2, a(n) is also the smallest absolute value of all negative values in row n of the triangle D(n, m) = n^2 - m^2 - 2*n*m, for 2 <= m + 1 <= n. The negative values in row n start with m = floor(n/(1 + sqrt(2))) + 1 = ceiling(n/(1 + sqrt(2))). See also a comment on A087056 for the smallest positive numbers in row n >= 3. - Wolfdieter Lang, Jun 11 2015
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LINKS
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FORMULA
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a(n) = A087058(n) - 2*n^2 = A087057(n)^2 - 2*n^2 = (1 + A001951(n))^2 - 2*n^2 = (1 + floor(n*sqrt(2)))^2 - 2*n^2.
a(n) = 2*c(n)^2 - (n - c(n))^2, with c(n) := ceiling(n/(1 + sqrt(2))), n >= 1. - Wolfdieter Lang, Jun 11 2015
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EXAMPLE
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a(10) = 25 because the difference between 2*10^2 = 200 and the next greater square number (225) is 25.
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MATHEMATICA
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(Floor[Sqrt[#]]+1)^2-#&/@Table[2n^2, {n, 80}] (* Harvey P. Dale, Jan 15 2023 *)
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PROG
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(PARI) a(n) = (1 + sqrtint(2*n^2))^2 - 2*n^2 \\ Michel Marcus, Jun 25 2013
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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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STATUS
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approved
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