A135708 -id:A135708 - cp's OEIS frontend

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

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

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

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

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).

Cf. A027434, A027709, A135708, A182834.

a078633 n = 2 * n + ceiling (2 * sqrt (fromIntegral n))
-- Reinhard Zumkeller, Aug 05 2014

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

a(n) = 2*n + ceil(2*sqrt(n)); \\ Michel Marcus, Mar 26 2018

from math import isqrt
def A078633(n): return (m:=n<<1)+1+isqrt((m<<1)-1) # Chai Wah Wu, Jul 28 2022

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

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

A135711 Minimal perimeter of a polyhex with n cells.

Original entry on oeis.org

6, 10, 12, 14, 16, 18, 18, 20, 22, 22, 24, 24, 26, 26, 28, 28, 30, 30, 30, 32, 32, 34, 34, 34, 36, 36, 36, 38, 38, 38, 40, 40, 40, 42, 42, 42, 42, 44, 44, 44, 46, 46, 46, 46, 48, 48, 48, 48, 50, 50, 50, 50, 52, 52, 52, 52, 54, 54, 54, 54, 54, 56, 56, 56, 56, 58, 58, 58, 58, 58, 60, 60

Offset: 1

Tanya Khovanova, Mar 04 2008

nonn

Y. S. Kupitz, "On the maximal number of appearances of the minimal distance among n points in the plane", in Intuitive geometry: Proceedings of the 3rd international conference held in Szeged, Hungary, 1991; Amsterdam: North-Holland: Colloq. Math. Soc. Janos Bolyai. 63, 217-244.

Michael De Vlieger, Table of n, a(n) for n = 1..10000
Li Gan, Stéphane Ouvry, and Alexios P. Polychronakos, Algebraic area enumeration of random walks on the honeycomb lattice, arXiv:2107.10851 [math-ph], 2021.
Greg Malen, Érika Roldán, and Rosemberg Toalá-Enríquez, Extremal {p, q}-Animals, Ann. Comb. (2023). See Corollary 1.9 at p. 8.

Cf. A000228 (number of hexagonal polyominoes (or planar polyhexes) with n cells), A135708.

Analogs for triangles, squares, cubes: A067628, A027709, A075777.

Table[2Ceiling[Sqrt[12n-3]],{n,120}] (* Harvey P. Dale, Dec 29 2019 *)

It is easy to use the formula of Harborth given in A135708 to show that a(n) = 2*ceiling(sqrt(12*n-3)). - Sascha Kurz, Mar 05 2008

2*A135708(n) - a(n) = 6n. - Tanya Khovanova, Mar 07 2008

More terms from N. J. A. Sloane, Mar 05 2008

A137228 Minimal total number of edges in a polyiamond consisting of n triangular cells.

Original entry on oeis.org

3, 5, 7, 9, 11, 12, 14, 16, 18, 19, 21, 23, 24, 26, 28, 29, 31, 33, 34, 36, 38, 39, 41, 42, 44, 46, 47, 49, 51, 52, 54, 55, 57, 59, 60, 62, 63, 65, 67, 68, 70, 71, 73, 75, 76, 78, 79, 81, 83, 84, 86, 87, 89, 90, 92, 94, 95, 97, 98, 100, 102, 103, 105, 106, 108, 109, 111, 113

Offset: 1

Tanya Khovanova, Mar 07 2008

nonn

Cf. A135708, A067628, A000577.

a(n) = (3n + A067628(n))/2.

A141135 Minimal number of unit edges required to construct n regular pentagons when allowing edge-sharing.

Original entry on oeis.org

5, 9, 13, 17, 21, 24, 28, 32, 36, 39, 43, 47, 50, 54, 58, 61, 65, 69, 72, 76, 80, 83, 87, 90, 94, 98, 101, 105, 109, 112

Offset: 1

Ralph H. Buchholz (teufel_pi(AT)yahoo.com), Jun 08 2008

			a(6) = 24 since the first pentagon requires 5 edges, the 2nd, 3rd, 4th and 5th pentagons require an additional 4 edges each and the 6th pentagon requires 3 edges since it can share 2 edges (if one tiles via a 6-cycle). Thus 24 = 5 + 4 + 4 + 4 + 4 + 3.

Ralph H. Buchholz, Spiral polygon series, preprint 1985 SMJ 31, School Mathematics Journal, 1995.
Ralph H. Buchholz and Warwick de Launey, Edge minimization, June 1996, (revised June 2008).
Ralph H. Buchholz and Warwick de Launey, An edge minimization problem for regular polygons, The Electronic Journal of Combinatorics, Volume 16, Issue 1 (2009), #R90.

Cf. equilateral triangles A137228, squares A078633, regular hexagons A135708.

Cf. A121149.

Conjectures from Colin Barker, Apr 05 2019: (Start)
G.f.: x*(5 + 4*x + 4*x^2 - x^3 - x^5 + x^8 - x^9) / ((1 - x)^2*(1 + x + x^2)).
a(n) = a(n-1) + a(n-3) - a(n-4) for n>10.
(End)
Conjecture: if n is a term in A121149, a(n) = a(n-1) + 3, otherwise a(n) = a(n-1) + 4. - Jinyuan Wang, Apr 05 2019

a(21)-a(30) from Jinyuan Wang, Apr 05 2019

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

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A135711 Minimal perimeter of a polyhex with n cells.

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A137228 Minimal total number of edges in a polyiamond consisting of n triangular cells.

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A141135 Minimal number of unit edges required to construct n regular pentagons when allowing edge-sharing.

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