|
%I #37 Jul 29 2023 05:05:04
%S 2,4,6,8,9,11,12,14,15,16,18,19,20,22,23,24,25,27,28,29,30,32,33,34,
%T 35,36,38,39,40,41,42,44,45,46,47,48,49,51,52,53,54,55,56,58,59,60,61,
%U 62,63,64,66,67,68,69,70,71,72,74,75,76,77,78,79,80,81,83,84,85,86,87,88,89,90,92
%N a(0)=2; for n > 0, a(n) = n + floor(sqrt(4n-3)) + 2.
%C 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]
%C For n > 0, a(n) = ceiling((1 + sqrt(n))^2). Empirical observation. - _Ronald S. Tiberio_, Jun 24 2018
%C 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
%C From _Ya-Ping Lu_, Mar 17 2022: (Start)
%C 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).
%C .
%C 37--36*-35*-34*-33*-32*-31
%C | |
%C 38* 17--16*-15*-14*-13 30*
%C | | | |
%C 39* 18* 5---4*--3 12* 29*
%C | | | | | |
%C 40* 19* 6* 1---2* 11* 28*
%C | | | | |
%C 41* 20* 7---8*--9*-10 27*
%C | | |
%C 42* 21--22*-23*-24*-25*-26
%C |
%C 43--44*-45*-46*-47*-48*-49* (End)
%H Robert Israel, <a href="/A080037/b080037.txt">Table of n, a(n) for n = 0..10000</a>
%H B. Cloitre, N. J. A. Sloane and M. J. Vandermast, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL6/Cloitre/cloitre2.html">Numerical analogues of Aronson's sequence</a>, J. Integer Seqs., Vol. 6 (2003), #03.2.2.
%H B. Cloitre, N. J. A. Sloane and M. J. Vandermast, <a href="https://arxiv.org/abs/math/0305308">Numerical analogues of Aronson's sequence</a>, arXiv:math/0305308 [math.NT], 2003.
%F 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
%p f:= n -> n + floor(sqrt(4*n-3)) + 2:
%p f(0):= 2:
%p map(f, [$0..100]); # _Robert Israel_, Jul 27 2023
%o (Python)
%o from math import isqrt
%o def A080037(n): return n+2+isqrt((n<<2)-1) if n else 2 # _Chai Wah Wu_, Jul 27 2022
%Y Cf. A000267, A080036, A033638 (complement, except 2).
%K nonn
%O 0,1
%A _N. J. A. Sloane_, Mar 14 2003
|
