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

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

Author

N. J. A. Sloane

Keywords

• nonn
• easy
• nice

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

Crossrefs

Cf. A000302 (subsequence of squares).

Essentially the same as A109965.

Programs

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

Mathematica

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

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

Extensions

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