cp's OEIS Frontend

A078633 Smallest number of sticks of length 1 needed to construct n squares with sides of length 1.

%I A078633 #49 Apr 21 2025 03:29:37
%S A078633 4,7,10,12,15,17,20,22,24,27,29,31,34,36,38,40,43,45,47,49,52,54,56,
%T A078633 58,60,63,65,67,69,71,74,76,78,80,82,84,87,89,91,93,95,97,100,102,104,
%U A078633 106,108,110,112,115,117,119,121,123,125,127,130,132,134,136,138,140,142
%N A078633 Smallest number of sticks of length 1 needed to construct n squares with sides of length 1.
%C A078633 A182834(a(n)) mod 2 = 0, or, where even terms occur in A182834. - _Reinhard Zumkeller_, Aug 05 2014
%H A078633 Reinhard Zumkeller, <a href="/A078633/b078633.txt">Table of n, a(n) for n = 1..10000</a>
%H A078633 Ralph H. Buchholz and Warwick De Launey, <a href="https://sites.google.com/site/teufelpi/papers/sth.pdf">The square, the triangle and the hexagon</a>, 1996.
%H A078633 Douglas A. Torrance, <a href="https://arxiv.org/abs/1906.01541">Enumeration of planar Tangles</a>, arXiv:1906.01541 [math.CO], 2019.
%H A078633 Douglas A. Torrance, <a href="https://doi.org/10.1007/s12044-020-00575-7">Enumeration of planar Tangles</a>, Proc. Math. Sci. 130 (50) (2020).
%F A078633 a(n) = 2*n + ceiling(2*sqrt(n)) = 2*n + A027434(n).
%F A078633 a(n) = (4*n + A027709(n))/2. - _Tanya Khovanova_, Mar 07 2008
%e A078633 a(2)=7 because we have following construction:
%e A078633    _ _
%e A078633   |_|_|
%t A078633 Table[2n+Ceiling[2Sqrt[n]],{n,70}] (* _Harvey P. Dale_, Jun 20 2011 *)
%o A078633 (Haskell)
%o A078633 a078633 n = 2 * n + ceiling (2 * sqrt (fromIntegral n))
%o A078633 -- _Reinhard Zumkeller_, Aug 05 2014
%o A078633 (PARI) a(n) = 2*n + ceil(2*sqrt(n)); \\ _Michel Marcus_, Mar 26 2018
%o A078633 (Python)
%o A078633 from math import isqrt
%o A078633 def A078633(n): return (m:=n<<1)+1+isqrt((m<<1)-1) # _Chai Wah Wu_, Jul 28 2022
%Y A078633 Cf. A027434, A027709, A135708, A182834.
%K A078633 nonn,easy,nice
%O A078633 1,1
%A A078633 Mambetov Timur and Takenov Nurdin (timur_teufel(AT)mail.ru), Dec 12 2002