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A184676 a(n) = n + floor((n/2-1/(4*n))^2); complement of A183867.

%I A184676 #39 Jul 28 2025 02:32:17
%S A184676 1,2,5,7,11,14,19,23,29,34,41,47,55,62,71,79,89,98,109,119,131,142,
%T A184676 155,167,181,194,209,223,239,254,271,287,305,322,341,359,379,398,419,
%U A184676 439,461,482,505,527,551,574,599,623,649,674,701,727,755,782,811,839
%N A184676 a(n) = n + floor((n/2-1/(4*n))^2); complement of A183867.
%C A184676 a(n) is also the number of trees with n vertices with diameter (n-3). - _Erich Friedman_, Apr 06 2017
%H A184676 Nathaniel Johnston, <a href="/A184676/b184676.txt">Table of n, a(n) for n = 1..5000</a>
%H A184676 <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (2,0,-2,1).
%F A184676 a(n) = n+floor((n/2-1/(4*n))^2).
%F A184676 a(n) = A198442(n+2)-1. - _Fred Daniel Kline_, Jun 24 2016
%F A184676 G.f.: x*(1 + x^2 - x^3)/((1 - x)^3*(1 + x)). - _Ilya Gutkovskiy_, Jun 24 2016
%p A184676 seq(n+floor((n/2-1/(4*n))^2), n=1..56); # _Nathaniel Johnston_, Jun 26 2011
%t A184676 a[n_]:=n+Floor[(n/2-1/(4n))^2];
%t A184676 b[n_]:=n+Floor[n^(1/2)+(n+1/2)^(1/2)];
%t A184676 Table[a[n],{n,1,120}]   (* A184676 *)
%t A184676 Table[b[n],{n,1,120}]   (* A183867 *)
%t A184676 FindLinearRecurrence[Table[a[n],{n,1,120}]]
%t A184676 LinearRecurrence[{2,0,-2,1},{1,2,5,7},56] (* _Ray Chandler_, Aug 02 2015 *)
%t A184676 Table[n + Floor[(n/2 - 1/(4 n))^2], {n, 60}] (* _Vincenzo Librandi_, Dec 09 2015 *)
%t A184676 Table[Ceiling[n/2] (2 + Ceiling[n/2] - Mod[n, 2]) - 1, {n, 1, 56}] (* _Fred Daniel Kline_, Jun 24 2016 *)
%o A184676 (PARI) a(n) = n+floor((n/2-1/(4*n))^2); \\ _Michel Marcus_, Dec 09 2015
%o A184676 (Magma) [n+Floor((n/2-1/(4*n))^2): n in [1..60]]; // _Vincenzo Librandi_, Dec 09 2015
%Y A184676 Cf. A183867, A184625, A198442.
%K A184676 nonn,easy
%O A184676 1,2
%A A184676 _Clark Kimberling_, Jan 19 2011