A078633 -id:A078633 - cp's OEIS frontend

A237347 First differences of A078633.

3, 3, 2, 3, 2, 3, 2, 2, 3, 2, 2, 3, 2, 2, 2, 3, 2, 2, 2, 3, 2, 2, 2, 2, 3, 2, 2, 2, 2, 3, 2, 2, 2, 2, 2, 3, 2, 2, 2, 2, 2, 3, 2, 2, 2, 2, 2, 2, 3, 2, 2, 2, 2, 2, 2, 3, 2, 2, 2, 2, 2, 2, 2, 3, 2, 2, 2, 2, 2, 2, 2, 3, 2, 2, 2, 2, 2, 2, 2, 2, 3, 2, 2, 2, 2, 2

Offset: 1

Reinhard Zumkeller, Mar 18 2014

nonn

a(n) = A078633(n+1) - A078633(n);
2 <= a(n) <= 3;
a(A049068(n)) = 2; a(A002620(n)) = 3;
A237347(n) = abs(A167752(n)) + 2.

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

a237347 n = a237347_list !! (n-1)
a237347_list = zipWith (-) (tail a078633_list) a078633_list

A182834 Complement of A007590, except for initial zeros.

Original entry on oeis.org

1, 3, 5, 6, 7, 9, 10, 11, 13, 14, 15, 16, 17, 19, 20, 21, 22, 23, 25, 26, 27, 28, 29, 30, 31, 33, 34, 35, 36, 37, 38, 39, 41, 42, 43, 44, 45, 46, 47, 48, 49, 51, 52, 53, 54, 55, 56, 57, 58, 59, 61, 62, 63, 64, 65, 66, 67, 68, 69, 70, 71, 73, 74, 75, 76, 77, 78, 79, 80, 81, 82, 83, 85, 86, 87, 88, 89, 90, 91, 92

Offset: 1

Clark Kimberling, Jan 07 2011

A245575(a(n)) mod 2 = 0, or, where even terms occur in A245575; a(A078633(n)) mod 2 = 0. - Reinhard Zumkeller, Aug 05 2014

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

Cf. A007590.

Cf. A000196, A078633, A245575.

a182834 n = a000196 (2 * n - 2) + n  -- Reinhard Zumkeller, Aug 05 2014

  a=2; b=-2;
  Table[n+Floor[(a*n+b)^(1/2)],{n,80}]
  Table[n-1+Ceiling[(n*n-b)/a],{n,50}]

from math import isqrt
def A182834(n): return n+isqrt(n-1<<1) # Chai Wah Wu, Jul 28 2022

a(n) = n + floor(sqrt(2n-2)).

A117227 Minimum number of unit segments required to construct n unit cubes.

Original entry on oeis.org

12, 20, 28, 33, 41, 46, 51, 54, 62, 67, 72, 75, 83, 88, 93, 96, 101, 104, 112, 117, 122, 125, 130, 133, 138, 141, 144

Offset: 1

Ron R. King, Apr 22 2006

This generalizes the 2-dimensional version which is A078633.

			a(2)=20 because 12 unit segments are required to construct each cube but 4 units are shared. I.e., a(2)=12+8=20.

Cf. A078633.

A331207 Minimum number of matchsticks sufficient to form the edges of exactly n squares of any size >= 1.

Original entry on oeis.org

0, 4, 7, 10, 13, 12, 15, 18, 17, 20, 23, 22, 25, 28, 24, 27, 30, 29, 32, 35, 31, 34, 37, 36, 39, 42, 38, 41, 44, 43, 40, 43, 46, 45

Offset: 0

John King, Jan 12 2020

John King, diagram for A331207 & A331208

Cf. A027709, A078633, A331208.

A331208 The minimum perimeter for exactly n matchstick squares of size >= 1.

Original entry on oeis.org

0, 4, 6, 8, 10, 8, 10, 12, 10, 12, 14, 12, 14, 16, 12, 14, 16, 14, 16, 18, 14, 16, 18, 16, 18, 20, 16, 18, 18, 18, 16, 18, 20

Offset: 0

John King, Jan 12 2020

John King, Graphic showing initial terms and growth for A331207 & A331208

Cf. A027709, A331207, A078633.

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

A275868 Numbers n tracing out a spiral path in a pentagonal Z module thereby creating a ten-fold twin pattern with relations to quasicrystals.

Original entry on oeis.org

0, 1, 2, 3, 2, 3, 4, 3, 4, 5, 4, 5, 4, 5, 6, 5, 6, 5, 6, 7, 6, 7, 6, 7, 6, 7, 8, 7, 8, 7, 8, 7, 8, 9, 8, 9, 8, 9, 8, 9, 8, 9, 0, 9, 0, 9, 0, 9, 0, 9, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 3, 2, 3, 2, 3, 2, 3, 2, 3, 2, 3, 2, 3, 4, 3, 4, 3, 4, 3, 4, 3, 4, 3, 4, 3

Offset: 1

Wolfgang Hornfeck, May 19 2017

Interpreted as consecutive steps along directions according to a basis of vectors represented by the tenth roots of unity in the complex number plane, the sequence traces out the path of a single spiral of a ten-fold twin pattern. All points are located on a pentagonal Z module (following the ideas of Quiquandon et al.). The ten-fold twin pattern is unique in that the local structure across the twin boundaries is identically coherent to the local structure within the twin domains. The ten-fold twin pattern is enantiomorphous, depending on the sign of the irrational shift of 1/(4*tau), with tau = (1+sqrt(5))/2 the Golden Ratio, along a [110] direction of the twin domain's orthorhombic unit cell. The sequence expresses the fact that the ten-fold twin pattern has no adjustable parameters, except for an arbitrary general scaling factor.

G. C. Greubel, Table of n, a(n) for n = 1..1000
W. Hornfeck, R. Kobold, M. Kolbe, D. Herlach, Quasicrystal nucleation in an intermetallic glass-former, arXiv:1410.2952 [cond-mat.mtrl-sci], 2014.
M. Quiquandon, D. Gratias, A. Sirindil, R. Portier, Merohedral twins revisited: quinary twins and beyond, Acta Cryst. A, 72 (2016), 55-61.

Cf. A001622, A078633, A172471.

Table[Mod[Floor[Sqrt[2*(i-1)]]+If[MemberQ[Table[2*j+Ceiling[2*Sqrt[j]],{j,1,i}],i],1,0],10],{i,1,100}]

a(n) = floor(sqrt( 2*(n-1) )) + [n in { 2*k + ceiling(2*sqrt(k)) | k in N}] mod 10. Note, that floor(sqrt( 2*n )) is A172471 (here corrected for its offset in the combined formula), while 2*k + ceiling(2*sqrt(k)) is A078633. [] denotes the Iverson bracket.

cp's OEIS Frontend

A237347 First differences of A078633.

Views

Author

Keywords

Comments

Links

Programs

Haskell

A182834 Complement of A007590, except for initial zeros.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Haskell

Mathematica

Python

Formula

A117227 Minimum number of unit segments required to construct n unit cubes.

Views

Author

Keywords

Comments

Examples

Crossrefs

A331207 Minimum number of matchsticks sufficient to form the edges of exactly n squares of any size >= 1.

Views

Author

Keywords

Links

Crossrefs

A331208 The minimum perimeter for exactly n matchstick squares of size >= 1.

Views

Author

Keywords

Links

Crossrefs

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

Views

Author

Keywords

Examples

Links

Crossrefs

Formula

Extensions

A275868 Numbers n tracing out a spiral path in a pentagonal Z module thereby creating a ten-fold twin pattern with relations to quasicrystals.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

Formula