OFFSET
0,1
COMMENTS
a(0)=2, a(1)=4; for n > 2, a(n) = a(n-1) + 1 if n is already in the sequence, a(n) = a(n-1) + 2 otherwise. [corrected by Jon E. Schoenfield, Jun 24 2018]
For n > 0, a(n) = ceiling((1 + sqrt(n))^2). Empirical observation. - Ronald S. Tiberio, Jun 24 2018
For n > 0, a(n) is the minimal number of thumbtacks needed to secure n square sheets of paper on a bulletin board - one thumbtack in each corner of each sheet - slight overlap allowed so that one thumbtack can secure up to four sheets. - Ronald S. Tiberio, Jun 24 2018
From Ya-Ping Lu, Mar 17 2022: (Start)
Integers m such that connecting the external lattice points of a square spiral with m lattice points forms a polyomino (m is indicated by * in the figure below).
.
37--36*-35*-34*-33*-32*-31
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38* 17--16*-15*-14*-13 30*
| | | |
39* 18* 5---4*--3 12* 29*
| | | | | |
40* 19* 6* 1---2* 11* 28*
| | | | |
41* 20* 7---8*--9*-10 27*
| | |
42* 21--22*-23*-24*-25*-26
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43--44*-45*-46*-47*-48*-49* (End)
LINKS
Robert Israel, Table of n, a(n) for n = 0..10000
B. Cloitre, N. J. A. Sloane and M. J. Vandermast, Numerical analogues of Aronson's sequence, J. Integer Seqs., Vol. 6 (2003), #03.2.2.
B. Cloitre, N. J. A. Sloane and M. J. Vandermast, Numerical analogues of Aronson's sequence, arXiv:math/0305308 [math.NT], 2003.
FORMULA
G.f.: (2-z)/(1-z)^2 + Sum_{k >= 1} z^(k^2+1)/(1-z) + Sum_{k >= 0} z^(k^2+k+1)/(1-z) = 1/2 + 1/(z-1)^2 - 1/(2*(z-1)) + z^(3/4)*JacobiTheta2(0,z)/(2*(1-z)) + z*JacobiTheta3(0,z)/(2*(1-z)). - Robert Israel, Jul 27 2023
MAPLE
PROG
(Python)
from math import isqrt
def A080037(n): return n+2+isqrt((n<<2)-1) if n else 2 # Chai Wah Wu, Jul 27 2022
CROSSREFS
KEYWORD
nonn
AUTHOR
N. J. A. Sloane, Mar 14 2003
STATUS
approved