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A050271
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Numbers k such that k = floor(sqrt(k)*ceiling(sqrt(k))).
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1
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1, 2, 3, 4, 7, 8, 9, 14, 15, 16, 23, 24, 25, 34, 35, 36, 47, 48, 49, 62, 63, 64, 79, 80, 81, 98, 99, 100, 119, 120, 121, 142, 143, 144, 167, 168, 169, 194, 195, 196, 223, 224, 225, 254, 255, 256, 287, 288, 289, 322, 323, 324, 359, 360, 361, 398, 399, 400
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OFFSET
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1,2
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COMMENTS
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Is a(n) asymptotic to C*n^(3/2) where 1/2 < C < 1?
Consists exactly of numbers of the forms j^2 - 2, j^2 - 1, and j^2. As such, is asymptotic to (1/9)*n^2. - Ivan Neretin, Feb 08 2017
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LINKS
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FORMULA
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a(n) = floor((n + 4)/3)^2 + ((n + 1) mod 3) - 2. - Ivan Neretin, Feb 08 2017
a(n) = a(n-1) + 2*a(n-3) - 2*a(n-4) - a(n-6) + a(n-7) for n > 7.
G.f.: x*(1 + x + x^2 - x^3 + x^4 - x^5) / ((1 - x)^3*(1 + x + x^2)^2).
(End)
Sum_{n>=1} 1/a(n) = 2 + Pi^2/6 - cot(sqrt(2)*Pi)*Pi/(2*sqrt(2)).
Sum_{n>=1} (-1)^(n+1)/a(n) = 1 + Pi^2/12 + cosec(sqrt(2)*Pi)*Pi/(2*sqrt(2)). (End)
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MAPLE
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a:=n->floor((n+4)/3)^2+irem(n+1, 3)-2:
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MATHEMATICA
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Select[Range@400, Floor[(r = Sqrt@#)*Ceiling@r] == # &] (* Ivan Neretin, Feb 08 2017 *)
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PROG
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(PARI) isok(n) = floor(sqrt(n)*ceil(sqrt(n))) == n; \\ Michel Marcus, Nov 22 2013
(PARI) Vec(x*(1 + x + x^2 - x^3 + x^4 - x^5) / ((1 - x)^3*(1 + x + x^2)^2) + O(x^100)) \\ Colin Barker, Feb 09 2017
(Python)
a, b = divmod(n+4, 3)
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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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STATUS
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approved
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