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

4, 7, 10, 12, 15, 17, 20, 22, 24, 27, 29, 31, 34, 36, 38, 40, 43, 45, 47, 49, 52, 54, 56, 58, 60, 63, 65, 67, 69, 71, 74, 76, 78, 80, 82, 84, 87, 89, 91, 93, 95, 97, 100, 102, 104, 106, 108, 110, 112, 115, 117, 119, 121, 123, 125, 127, 130, 132, 134, 136, 138, 140, 142

Offset

1,1

Comments

A182834(a(n)) mod 2 = 0, or, where even terms occur in A182834. - Reinhard Zumkeller, Aug 05 2014

Links

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000

Ralph H. Buchholz and Warwick De Launey, The square, the triangle and the hexagon, 1996.

Douglas A. Torrance, Enumeration of planar Tangles, arXiv:1906.01541 [math.CO], 2019.

Douglas A. Torrance, Enumeration of planar Tangles, Proc. Math. Sci. 130 (50 (2020)

Formula

a(n) = 2*n + ceiling(2*sqrt(n)) = 2*n + A027434(n).

a(n) = (4*n + A027709(n))/2. - Tanya Khovanova, Mar 07 2008

Example

a(2)=7 because we have following construction:
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Mathematica

Table[2n+Ceiling[2Sqrt[n]], {n, 70}] (* Harvey P. Dale, Jun 20 2011 *)

Prog

(Haskell)
a078633 n = 2 * n + ceiling (2 * sqrt (fromIntegral n))
-- Reinhard Zumkeller, Aug 05 2014
(PARI) a(n) = 2*n + ceil(2*sqrt(n)); \\ Michel Marcus, Mar 26 2018
(Python)
from math import isqrt
def A078633(n): return (m:=n<<1)+1+isqrt((m<<1)-1) # Chai Wah Wu, Jul 28 2022

Crossrefs

Cf. A027434, A027709, A135708, A182834.

Sequence in context: A310675 A176292 A050173 * A190008 A184911 A103762

Adjacent sequences: A078630 A078631 A078632 * A078634 A078635 A078636

Keyword

nonn,easy,nice

Author

Mambetov Timur and Takenov Nurdin (timur_teufel(AT)mail.ru), Dec 12 2002

Status

approved