A002984 a(0) = 1; for n > 0, a(n) = a(n-1) + floor(sqrt(a(n-1))). (Formerly M0554)

1, 2, 3, 4, 6, 8, 10, 13, 16, 20, 24, 28, 33, 38, 44, 50, 57, 64, 72, 80, 88, 97, 106, 116, 126, 137, 148, 160, 172, 185, 198, 212, 226, 241, 256, 272, 288, 304, 321, 338, 356, 374, 393, 412, 432, 452, 473, 494, 516, 538, 561, 584, 608, 632, 657, 682, 708, 734

Offset

0,2

Comments

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

This sequence contains infinitely many squares. - Philippe Deléham, Apr 03 2009

The squares in this sequence are precisely the powers of 4. - Franklin T. Adams-Watters, Jan 06 2014

References

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Links

T. D. Noe, Table of n, a(n) for n = 0..1000

Formula

a(n+1) = a(n) + A000196(a(n)). - Reinhard Zumkeller, Dec 28 2011

Conjecture: a(n) ~ n^2/4. - José María Grau Ribas, Feb 13 2024

Mathematica

NestList[ # + Floor[ Sqrt[ # ] ] &, 1, 50 ]

Prog

(Haskell)
a002984 n = a002984_list !! n
a002984_list = iterate (\x -> x + a000196 x) 1
-- Reinhard Zumkeller, Dec 28 2011
(Magma) [n le 0 select 1 else Self(n)+Floor(Sqrt(Self(n))): n in [0..60]]; // Bruno Berselli, Feb 15 2013

Crossrefs

Cf. A000302 (subsequence of squares).

Essentially the same as A109965.

Sequence in context: A008748 A089649 A049700 * A109965 A008669 A055104

Adjacent sequences: A002981 A002982 A002983 * A002985 A002986 A002987

Keyword

nonn,easy,nice

Author

N. J. A. Sloane

Extensions

More terms from Larry Reeves (larryr(AT)acm.org), Dec 14 2000

Status

approved