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A002984 a(0) = 1; for n > 0, a(n) = a(n-1) + floor(sqrt(a(n-1))).

%I A002984 M0554 #57 Feb 14 2024 15:03:34
%S A002984 1,2,3,4,6,8,10,13,16,20,24,28,33,38,44,50,57,64,72,80,88,97,106,116,
%T A002984 126,137,148,160,172,185,198,212,226,241,256,272,288,304,321,338,356,
%U A002984 374,393,412,432,452,473,494,516,538,561,584,608,632,657,682,708,734
%N A002984 a(0) = 1; for n > 0, a(n) = a(n-1) + floor(sqrt(a(n-1))).
%C A002984 For n > 3 we have a(n) < n^2/4; for n > 44 we have a(n) > n^2/5. - _Stefan Steinerberger_, Apr 17 2006
%C A002984 This sequence contains infinitely many squares. - _Philippe Deléham_, Apr 03 2009
%C A002984 The squares in this sequence are precisely the powers of 4. - _Franklin T. Adams-Watters_, Jan 06 2014
%D A002984 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%H A002984 T. D. Noe, <a href="/A002984/b002984.txt">Table of n, a(n) for n = 0..1000</a>
%F A002984 a(n+1) = a(n) + A000196(a(n)). - _Reinhard Zumkeller_, Dec 28 2011
%F A002984 Conjecture: a(n) ~ n^2/4. - _José María Grau Ribas_, Feb 13 2024
%t A002984 NestList[ # + Floor[ Sqrt[ # ] ] &, 1, 50 ]
%o A002984 (Haskell)
%o A002984 a002984 n = a002984_list !! n
%o A002984 a002984_list = iterate (\x -> x + a000196 x) 1
%o A002984 -- _Reinhard Zumkeller_, Dec 28 2011
%o A002984 (Magma) [n le 0 select 1 else Self(n)+Floor(Sqrt(Self(n))): n in [0..60]]; // _Bruno Berselli_, Feb 15 2013
%Y A002984 Cf. A000302 (subsequence of squares).
%Y A002984 Essentially the same as A109965.
%K A002984 nonn,easy,nice
%O A002984 0,2
%A A002984 _N. J. A. Sloane_
%E A002984 More terms from Larry Reeves (larryr(AT)acm.org), Dec 14 2000