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A050271 Numbers n such that n = floor(sqrt(n)*ceiling(sqrt(n))). 1

%I

%S 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,

%T 98,99,100,119,120,121,142,143,144,167,168,169,194,195,196,223,224,

%U 225,254,255,256,287,288,289,322,323,324,359,360,361,398,399,400

%N Numbers n such that n = floor(sqrt(n)*ceiling(sqrt(n))).

%C Is a(n) asymptotic to C*n^(3/2) where 1/2 < C < 1?

%C Consists exactly of numbers of the forms k^2 - 2, k^2 - 1, and k^2. As such, is asymptotic to 1/9 * n^2. - _Ivan Neretin_, Feb 08 2017

%H Ivan Neretin, <a href="/A050271/b050271.txt">Table of n, a(n) for n = 1..10000</a>

%H <a href="/index/Rec#order_07">Index entries for linear recurrences with constant coefficients</a>, signature (1,0,2,-2,0,-1,1).

%F a(n) = floor((n + 4)/3)^2 + ((n + 1) mod 3) - 2. - _Ivan Neretin_, Feb 08 2017

%F From _Colin Barker_, Feb 09 2017: (Start)

%F a(n) = a(n-1) + 2*a(n-3) - 2*a(n-4) - a(n-6) + a(n-7) for n>7.

%F G.f.: x*(1 + x + x^2 - x^3 + x^4 - x^5) / ((1 - x)^3*(1 + x + x^2)^2).

%F (End)

%t Select[Range@400, Floor[(r = Sqrt@#)*Ceiling@r] == # &] (* _Ivan Neretin_, Feb 08 2017 *)

%o (PARI) isok(n) = floor(sqrt(n)*ceil(sqrt(n))) == n; \\ _Michel Marcus_, Nov 22 2013

%o (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

%Y Cf. A000290, A005563, A008865.

%K nonn,easy

%O 1,2

%A _Benoit Cloitre_, May 10 2003

%E Data corrected by _Michel Marcus_ and _Benoit Cloitre_, Nov 22 2013

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Last modified May 7 10:18 EDT 2021. Contains 343650 sequences. (Running on oeis4.)