A027709 -id:A027709 - cp's OEIS frontend

A063655 Smallest semiperimeter of integral rectangle with area n.

2, 3, 4, 4, 6, 5, 8, 6, 6, 7, 12, 7, 14, 9, 8, 8, 18, 9, 20, 9, 10, 13, 24, 10, 10, 15, 12, 11, 30, 11, 32, 12, 14, 19, 12, 12, 38, 21, 16, 13, 42, 13, 44, 15, 14, 25, 48, 14, 14, 15, 20, 17, 54, 15, 16, 15, 22, 31, 60, 16, 62, 33, 16, 16, 18, 17, 68, 21, 26

Offset: 1

Floor van Lamoen, Jul 24 2001

nonn

Similar to A027709, which is minimal perimeter of polyomino of n cells, or equivalently, minimal perimeter of rectangle of area at least n and with integer sides. Present sequence is minimal semiperimeter of rectangle with area exactly n and with integer sides. - Winston C. Yang (winston(AT)cs.wisc.edu), Feb 03 2002
Semiperimeter b+d, d >= b, of squarest (smallest d-b) integral rectangle with area bd = n. That is, b = largest divisor of n <= sqrt(n), d = smallest divisor of n >= sqrt(n). a(n) = n+1 iff n is noncomposite (1 or prime). - Daniel Forgues, Nov 22 2009
From Juhani Heino, Feb 05 2019: (Start)
Basis for any thickness "frames" around the minimal area. Perimeter can be thought as the 0-thick frame, it is obviously 2a(n). Thickness 1 is achieved by laying unit tiles around the area, there are 2(a(n)+2) of them. Thickness 2 comes from the second such layer, now there are 4(a(n)+4) and so on. They all depend only on a(n), so they share this structure:
Every n > 1 is included. (For different thicknesses, every integer that can be derived from these with the respective formula. So, the perimeter has every even n > 2.)
For each square n > 1, a(n) = a(n-1).
a(1), a(2) and a(6) are the only unique values - the others appear multiple times.
(End)
Gives a discrete Uncertainty Principle. A complex function on an abelian group of order n and its Discrete Fourier Transform must have at least a(n) nonzero entries between them. This bound is achieved by the indicator function on a subgroup of size closest to sqrt(n). - Oscar Cunningham, Oct 10 2021
Also two times the median divisor of n, where the median of a multiset is either the middle part (for odd length), or the average of the two middle parts (for even length). The version for mean instead of median is A057020/A057021. Other doubled medians of multisets are: A360005 (prime indices), A360457 (distinct prime indices), A360458 (distinct prime factors), A360459 (prime factors), A360460 (prime multiplicities), A360555 (0-prepended differences). - Gus Wiseman, Mar 18 2023

			Since 15 = 1*15 = 3*5 and the 3*5 rectangle gives smallest semiperimeter 8, we have a(15)=8.

T. D. Noe, Table of n, a(n) for n = 1..10000
Roy Meshulam, An uncertainty inequality for finite abelian groups, arXiv:math/0312407 [math.CO], 2003.
Roy Meshulam, An uncertainty inequality for finite abelian groups, European Journal of Combinatorics, 27 (2006) 63-67.

Cf. A027709, A033676, A033677, A092510, A162348.
Positions of odd terms are A139710.
Positions of even terms are A139711.
A000005 counts divisors, listed by A027750.
A000975 counts subsets with integer median.
Cf. A003601, A026424, A057020/A057021, A325347, A359893, A359901.

A063655 := proc(n)
    local i,j;
    for i from floor(sqrt(n)) to 1 by -1 do
        j := floor(n/i) ;
        if i*j = n then
            return i+j;
        end if;
    end do:
end proc:
seq(A063655(n), n=1..80); # Winston C. Yang, Feb 03 2002

Table[d = Divisors[n]; len = Length[d]; If[OddQ[len], 2*Sqrt[n], d[[len/2]] + d[[1 + len/2]]], {n, 100}] (* T. D. Noe, Mar 06 2012 *)
Table[2*Median[Divisors[n]],{n,100}] (* Gus Wiseman, Mar 18 2023 *)

A063655(n) = { my(c=1); fordiv(n,d,if((d*d)>=n,if((d*d)==n,return(2*d),return(c+d))); c=d); (0); }; \\ Antti Karttunen, Oct 20 2017

from sympy import divisors
def A063655(n):
    d = divisors(n)
    l = len(d)
    return d[(l-1)//2] + d[l//2] # Chai Wah Wu, Jun 14 2019

a(n) = A033676(n) + A033677(n).
a(n) = A162348(2n-1) + A162348(2n). - Daniel Forgues, Sep 29 2014
a(n) = Min_{d|n} (n/d + d). - Ridouane Oudra, Mar 17 2024

Corrected and extended by Larry Reeves (larryr(AT)acm.org) and Dean Hickerson, Jul 26 2001

A027434 a(1) = 2; then defined by property that a(n) = smallest number >= a(n-1) such that successive runs have lengths 1,1,2,2,3,3,4,4.

Original entry on oeis.org

2, 3, 4, 4, 5, 5, 6, 6, 6, 7, 7, 7, 8, 8, 8, 8, 9, 9, 9, 9, 10, 10, 10, 10, 10, 11, 11, 11, 11, 11, 12, 12, 12, 12, 12, 12, 13, 13, 13, 13, 13, 13, 14, 14, 14, 14, 14, 14, 14, 15, 15, 15, 15, 15, 15, 15, 16, 16, 16, 16, 16, 16, 16, 16, 17, 17, 17, 17, 17, 17, 17, 17, 18, 18, 18, 18

Offset: 1

Sam Speed (SPEEDS(AT)msci.memphis.edu)

Also the sequence of first skipped terms for Beatty sequences in the family alpha = 1+sqrt(n)-sqrt(n-1). - Alisa Ediger, Jul 20 2016
Optimal cost for one-dimensional Racetrack over a distance n. - Jason Schoeters, Aug 18 2021
If b > 0 and c > 0 are the integer coefficients of a monic quadratic x^2 + b*x + c, it has integer roots if its discriminant d^2 = b^2 - 4c is a perfect square. This sequence is the values of b for increasing b sorted by b then c. The first pair of (b, c) = (2, 1) and has d = A082375(0) = 0. The n-th pair of (b, c) = (a(n), A350634(n)) and has d = A082375(n-1). - Frank M Jackson, Jan 21 2024

Sam Speed, An integer sequence (preprint).

William A. Tedeschi, Table of n, a(n) for n = 1..10000
A. Casteigts, M. Raffinot and J. Schoeters, VectorTSP: A Traveling Salesperson Problem with Racetrack-like acceleration constraints, Lemma 7, arXiv:2006.03666 [cs.DS], 2020-2021.

Cf. A000267, A049068, A027709.

Cf. A082375, A350634.

a027434 = (+ 1) . a000196 . (subtract 3) . (* 4)
a027434_list = 2 : concat (map (\x -> replicate (x `div` 2) x) [3..])
-- Reinhard Zumkeller, Mar 23 2013, Nov 22 2011

A027434:=n->ceil(2*sqrt(n)); seq(A027434(n), n=1..100); # Wesley Ivan Hurt, Mar 01 2014

Table[Ceiling[2*Sqrt[n]], {n, 100}] (* Wesley Ivan Hurt, Mar 01 2014 *)
Sort[Flatten[Table[#,{#[[1]]/2}]]]&/@Partition[Range[2,20],2]//Flatten (* Harvey P. Dale, Sep 05 2019 *)
lst = {}; Do[If[IntegerQ[d=Sqrt[b^2-4 c]], AppendTo[lst, b]], {b, 1, 20}, {c, 1, b^2/4}]; lst (* Frank M Jackson, Jan 21 2024 *)

a(n)=sqrtint(4*n-3)+1 \\ Charles R Greathouse IV, Feb 07 2012

from math import isqrt
def A027434(n): return 1+isqrt((n<<2)-1) # Chai Wah Wu, Jul 27 2022

a(n) = 1 + floor( sqrt(4*n-3) ) = 1+A000267(n-1).
a(n) = A049068(n) - n.
a(n) = A027709(n)/2. - Tanya Khovanova, Mar 04 2008
a(n) = ceiling(2*sqrt(n)). [Mircea Merca, Feb 07 2012]
a(n) = floor(1+sqrt(n)+sqrt(n-1)). - Alisa Ediger, Jul 20 2016
G.f.: x*(1 + x^(-1/4)*theta_2(x) + theta_3(x))/(2*(1 - x)), where theta_k(x) is the Jacobi theta function. - Ilya Gutkovskiy, Jul 20 2016
a(n) = 1 + floor(sqrt(4*n-1)). - Chai Wah Wu, Jul 27 2022
a(n) = sqrt((A082375(n))^2 + 4*A350634(n+1)). - Frank M Jackson, Jan 21 2024

More terms from Courtney Clipp (cclipp(AT)ashland.edu), Dec 08 2004

A067628 Minimal perimeter of polyiamond with n triangles.

Original entry on oeis.org

0, 3, 4, 5, 6, 7, 6, 7, 8, 9, 8, 9, 10, 9, 10, 11, 10, 11, 12, 11, 12, 13, 12, 13, 12, 13, 14, 13, 14, 15, 14, 15, 14, 15, 16, 15, 16, 15, 16, 17, 16, 17, 16, 17, 18, 17, 18, 17, 18, 19, 18, 19, 18, 19, 18, 19, 20, 19, 20, 19, 20, 21, 20, 21, 20, 21, 20, 21, 22, 21, 22, 21, 22

Offset: 0

Winston C. Yang (winston(AT)cs.wisc.edu), Feb 02 2002

nonn

A polyiamond is a shape made up of n congruent equilateral triangles.

Frank Harary and Heiko Harborth, Extremal animals, J. Combinatorics Information Syst. Sci., 1(1):1-8, 1976.

Stefano Spezia, Table of n, a(n) for n = 0..10000
Greg Malen, Érika Roldán, and Rosemberg Toalá-Enríquez, Extremal {p, q}-Animals, Ann. Comb. (2023). See Corollary 1.9 at p. 8.
Greg Malen and Érika Roldán, Polyiamonds Attaining Extremal Topological Properties, arXiv:1906.08447 [math.CO], 2019.
J. Yackel, R. R. Meyer, and I. Christou, Minimum-perimeter domain assignment, Mathematical Programming, vol. 78 (1997), pp. 283-303.
W. C. Yang and R. R. Meyer, Maximal and minimal polyiamonds, 2002.

Cf. A000105, A000577, A027709 (squares), A057729, A135711.

interface(quiet=true); for n from 0 to 100 do if (1 = 1) then temp1 := ceil(sqrt(6*n)); end if; if ((temp1 mod 2) = (n mod 2)) then temp2 := 0; else temp2 := 1; end if; printf("%d,", temp1 + temp2); od;

a(n)=2*ceil((n+sqrt(6*n))/2)-n; \\ Stefano Spezia, Oct 02 2019

from math import isqrt
def A067628(n): return (c:=isqrt(6*n-1)+1)+((c^n)&1) if n else 0 # Chai Wah Wu, Jul 28 2022

Let c(n) = ceiling(sqrt(6n)). Then a(n) is whichever of c(n) or c(n) + 1 has the same parity as n.

a(n) = 2*ceiling((n + sqrt(6*n))/2) - n (Harary and Harborth, 1976). - Stefano Spezia, Oct 02 2019

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

Original entry on oeis.org

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

A083479 The natural numbers with all terms of A033638 inserted.

Original entry on oeis.org

0, 1, 1, 1, 2, 2, 3, 3, 4, 5, 5, 6, 7, 7, 8, 9, 10, 10, 11, 12, 13, 13, 14, 15, 16, 17, 17, 18, 19, 20, 21, 21, 22, 23, 24, 25, 26, 26, 27, 28, 29, 30, 31, 31, 32, 33, 34, 35, 36, 37, 37, 38, 39, 40, 41, 42, 43, 43, 44, 45, 46, 47, 48, 49, 50, 50, 51, 52, 53, 54, 55, 56, 57, 57

Offset: 0

Alford Arnold, Jun 08 2003

Row n of A049597 has a(n+1) nonzero values.
When considering the set of nested parabolas defined by -(x^2) + p*x for integer values of p, a(n) tells us how many parabolas are intersected by the line from (1,n) to (n,n). - Gregory R. Bryant, Apr 01 2013
Number of distinct perimeters for polyominoes with n square cells. - Wesley Prosser, Sep 06 2017

			There are three 1's, one from the natural numbers and two from A033638.
When viewed as an array the sequence begins:
   0
   1
   1  1
   2  2
   3  3  4
   5  5  6
   7  7  8  9
  10 10 11 12
  13 13 14 15 16
  17 17 18 19 20
  21 21 22 23 24 25
  26 26 27 28 29 30
  ...

Reinhard Zumkeller, Table of n, a(n) for n = 0..10000

Cf. A002620, A027434, A027709, A033638, A049068, A049597.

Cf. A054243, A060510, A080037, A083480, A083906.

a083479 n = a083479_list !! n
a083479_list = m [0..] a033638_list where
   m xs'@(x:xs) ys'@(y:ys) | x <= y    = x : m xs ys'
                           | otherwise = y : m xs' ys
-- Reinhard Zumkeller, Apr 06 2012

[n eq 0 select 0 else (n+2)-Ceiling(Sqrt(4*n)): n in [0..100]]; // G. C. Greubel, Feb 17 2024

Table[(n + 2) - Ceiling@ Sqrt[4 n] - 2 Boole[n == 0], {n, 0, 73}] (* Michael De Vlieger, Sep 05 2017 *)

a(n):=((n+2)-ceiling(sqrt(4*n))); /* Gregory R. Bryant, Apr 01 2013 */

from math import isqrt
def A083479(n): return n+1-isqrt((n<<2)-1) if n else 0 # Chai Wah Wu, Jul 28 2022

[(n+2)-ceil(sqrt(4*n)) -2*int(n==0) for n in range(101)] # G. C. Greubel, Feb 17 2024

a(n) = (n+2) - ceiling(sqrt(4*n)), for n > 0. - Gregory R. Bryant, Apr 01 2013
From Wesley Prosser, Sep 06 2017: (Start)
a(n) = (n+2) - A027709(n)/2.
a(n) = (n+2) - A027434(n).
a(n) = (2n+2) - A049068(n).
a(n) = (2n+3) - A080037(n).
(End)

Edited and extended by David Wasserman, Nov 16 2004

A123663 Number of shared edges in a spiral of n unit squares.

Original entry on oeis.org

0, 1, 2, 4, 5, 7, 8, 10, 12, 13, 15, 17, 18, 20, 22, 24, 25, 27, 29, 31, 32, 34, 36, 38, 40, 41, 43, 45, 47, 49, 50, 52, 54, 56, 58, 60, 61, 63, 65, 67, 69, 71, 72, 74, 76, 78, 80, 82, 84, 85, 87, 89, 91, 93, 95, 97, 98, 100, 102, 104, 106, 108, 110, 112, 113, 115, 117, 119

Offset: 1

Zacariaz Martinez, Nov 15 2006

If one constructs a square (square 1) and then draws another square of identical size beside it (square 2), the squares share 1 edge. If one then places an identical square above square 2 (instead of continuing in a straight path), there are now 2 shared edges. Continuing this pattern in an outward spiral, one finds that the number of shared edges is 4, 5, 7, ...
Numbers a(n) such that a(n+1) = a(n) + 1 are (except for the leading zero) A074148. Otherwise a(n+1) = a(n) + 2. - Franklin T. Adams-Watters, Oct 17 2014.
This sequence is also the maximal number of shared edges among all polyominoes with n square cells.  This is the result of Harary and Harborth cited in the references. Once this is known the formula 2n - ceiling(2*sqrt(n)) comes from geometrical considerations and A027709. Namely, the 4n sides of the n squares making up the polyomino form the perimeter and come together in pairs along shared edges. Hence, 4n = perimeter + 2*shared edges. Maximizing shared edges minimizes perimeter and so maximum shared edges = (4n - minimum perimeter)/2 = (4n - 2ceiling(2*sqrt(n)))/2 = 2n - ceiling(2*sqrt(n)). This interpretation is important to landscape ecologists and is called the aggregation index in the GIS program FRAGSTATS. - Julian F. Fleron, Nov 29 2016
a(n) is also the maximum degree of the cover graphs of lattice quotients of lattice congruences of the weak order on the symmetric group S_n. See Table 1 in the Hoang/Mütze reference in the Links section. - Torsten Muetze, Nov 28 2019
a(n) is also the number of pixels in H_{n-1}, where H_n (a pixelated piece of hyperbola x*y = n) is the set of the (x, y), ordered pairs of positive integers, such that x*y = n or (x*y < n and ((x+1)*y > n or x*(y+1) > n)). - Luc Rousseau, Dec 28 2019

F. Harary and H. Harborth, Extremal Animals, Journal of Combinatorics, Information & System Sciences, Vol. 1, No 1, 1-8 (1976).

Peter Kagey, Table of n, a(n) for n = 1..10000
Richard A. Brualdi and Geir Dahl, Frobenius-König theorem for classes of (0,+/-1)-matrices, Disc. Math. (2024) Vol. 347, Issue 6, 113951.
Hung Phuc Hoang and Torsten Mütze, Combinatorial generation via permutation languages. II. Lattice congruences, arXiv:1911.12078 [math.CO], 2019.

Cf. A002620.

A[1]:= 0:
for n from 2 to 100 do
  if issqr(2*A[n-1]+1) or issqr(2*A[n-1]+2) then A[n]:= A[n-1]+1
  else A[n]:= A[n-1]+2
  fi
od:
seq(A[n],n=1..100); # Robert Israel, Oct 21 2014

FoldList[Plus, 0, t = Table[2, {72}]; t[[ Table[ Ceiling[n/2] Floor[n/2], {n, 2, 16}] ]]--; t] (* Robert G. Wilson v, Jan 19 2007 *)

a(n)=2*n - sqrtint(4*n-1) - 1 \\ Charles R Greathouse IV, Nov 29 2016

from math import isqrt
def A123663(n): return (m:=n<<1)-1-isqrt((m<<1)-1) # Chai Wah Wu, Jul 28 2022

a123663 = [0]; k = 0; a_n = 0; (1..N).to_a.each{ |i| 2.times{ k.times{ a_n += 2; a123663 << a_n }; a_n += 1; a123663 << a_n; }; k += 1}

a(n) = 2n - ceiling(2*sqrt(n)). - Julian F. Fleron, Nov 29 2016

a(n) = a(n-1) + 2 - [n-1 is a square or a pronic number], where [] stands for the Iverson bracket. - Luc Rousseau, Dec 28 2019

Extended by Robert G. Wilson v, Jan 19 2007

A235382 a(n) = smallest number of unit squares required to enclose n units of area.

Original entry on oeis.org

4, 8, 10, 12, 12, 14, 14, 16, 16, 16, 18, 18, 18, 20, 20, 20, 20, 22, 22, 22, 22, 24, 24, 24, 24, 24, 26, 26, 26, 26, 26, 28, 28, 28, 28, 28, 28, 30, 30, 30, 30, 30, 30, 32, 32, 32, 32, 32, 32, 32, 34, 34, 34, 34, 34, 34, 34, 36, 36, 36, 36, 36

Offset: 0

L. Edson Jeffery, Jan 08 2014

Result attributed to the students Daring, et al., in the links section.

Charles R Greathouse IV, Table of n, a(n) for n = 0..10000
E. Daring, I. Guadarrama, S. Sprague, and C. Winterer, WhaleConjecture.
E. Daring, I. Guadarrama, S. Sprague, and C. Winterer, The Whale Theorem. (PDF contains incomplete proof.)
Kival Ngaokrajang, Illustration of initial terms

Cf. A027434, A027709, A232091.

[2*Ceiling(2*Sqrt(n))+4: n in [0..70]]; // Vincenzo Librandi, Jul 27 2016

Table[2 Ceiling[2 Sqrt[n]] + 4, {n, 0, 49}] (* Michael De Vlieger, Jul 21 2016 *)

a(n)=if(n,2*sqrtint(4*n-1)+6,4) \\ Charles R Greathouse IV, Jan 09 2014

from math import isqrt
def A235382(n): return 3+isqrt((n<<2)-1)<<1 if n else 4 # Chai Wah Wu, Jul 28 2022

a(n) = 2*ceiling(2*sqrt(n)) + 4.
a(n) = A027709(n) + 4.
a(n) = 2*A027434(n) + 4, n>0.

A342243 Triangle T(n,p) read by rows: the number of n-celled polyominoes with perimeter 2p, 2 <= p <= 1+n.

Original entry on oeis.org

1, 0, 1, 0, 0, 2, 0, 0, 1, 4, 0, 0, 0, 1, 11, 0, 0, 0, 1, 7, 27, 0, 0, 0, 0, 4, 21, 83, 0, 0, 0, 0, 2, 21, 91, 255, 0, 0, 0, 0, 1, 9, 89, 339, 847, 0, 0, 0, 0, 0, 6, 67, 393, 1360, 2829, 0, 0, 0, 0, 0, 1, 45, 325, 1713, 5255, 9734, 0, 0, 0, 0, 0, 1, 23, 275

Offset: 1

R. J. Mathar, Mar 07 2021

			The triangle has rows n=1,2,3,... and columns p=2,3,4,5,...:
  1;
  0, 1;
  0, 0, 2;
  0, 0, 1, 4;
  0, 0, 0, 1, 11;
  0, 0, 0, 1,  7, 27;
  0, 0, 0, 0,  4, 21, 83;
  0, 0, 0, 0,  2, 21, 91, 255;
  0, 0, 0, 0,  1,  9, 89, 339,  847;
  0, 0, 0, 0,  0,  6, 67, 393, 1360, 2829;
  0, 0, 0, 0,  0,  1, 45, 325, 1713, 5255, 9734;
  ...

John Mason, Table of T(n,k) for n <= 18 (n <= 17 from R. J. Mathar)
Andrew Clarke, Isoperimetrical Polyominoes
John Mason, Incomplete rows for n>18

Cf. A000105 (row sums), A057730 (column sums), A131482 (diagonal), A131487 (skew antidiagonal sums), A027709 (number of leading zeros per row), A100092 (first nonzero in each row).

A131487(e) = Sum_{e=2*n+p} T(n,p).

A100092 Number of n-celled polyominoes with minimum perimeter.

Original entry on oeis.org

1, 1, 1, 2, 1, 1, 1, 4, 2, 1, 6, 1, 1, 11, 4, 2, 1, 11, 6, 1, 1, 28, 11, 4, 2, 1, 35, 11, 6, 1, 1, 65, 28, 11, 4, 2, 1, 73, 35, 11, 6, 1, 1, 147, 65, 28, 11, 4, 2, 1, 182, 73, 35, 11, 6, 1, 1, 321, 147, 65, 28, 11, 4, 2, 1, 374, 182, 73, 35, 11, 6, 1, 1, 678, 321, 147, 65, 28, 11, 4, 2, 1

Offset: 0

Sascha Kurz, Nov 03 2004

nonn

An n-celled polyomino has minimum perimeter A027709(n) = 2*ceiling(2*sqrt(n)). - Dmitry Kamenetsky, Feb 27 2017

			a(9) = 1 because the 3 X 3 square is the unique polyomino with minimum perimeter.

Joerg Arndt, Table of n, a(n) for n = 0..144
Sascha Kurz, Counting polyominoes with minimum perimeter, submitted to Ars Combinatoria
Sascha Kurz, Counting polyominoes with minimum perimeter, arXiv:math/0506428 [math.CO], 2005-2015.
Kival Ngaokrajang, Illustration of initial terms ["A275966" should be changed to "A100092"]
Piotr Pikul, Tightest Arrangements of All Different Nets of a Cube, Math. Mag. (2024).

Cf. A027709, A100093, A100094, left nonzero term in row n of A342243.

(* Warning: some local maxima are precomputed from A100094. *)
A100094 = {2, 4, 11, 28, 65, 147, 321, 678, 1382, 2738, 5289 (* extend if needed *)};
amax = Last[A100094]; nmax = 144;
S[x_] := 1 + Sum[ x^(2*n + 1)*Product[ (x^(2*k - 1) - 1), {k, n}], {n, 0, nmax}] + O[x]^nmax;
A[x_] = Product[1/(1 - x^k), {k, 1, nmax}] + O[x]^nmax // Normal;
R[x_] := 1/4 (A[x]^4 + 3A[x^2]^2) + O[x]^nmax;
Q[x_] := 1/8 (A[x]^4 + 3A[x^2]^2 + 2S[x]^2 A[x^2] + 2A[x^4]) + O[x]^nmax;
r[k_] := SeriesCoefficient[R[x], {x, 0, k}];
q[k_] := SeriesCoefficient[Q[x], {x, 0, k}];
e[n_] := Module[{s, w}, s = Floor[Sqrt[n]]; a94Q[k_] := IntegerQ[w = Sqrt[k + n] - k] && w > 0; Which[Evaluate[Sequence @@ Flatten[Table[{a94Q[k], A100094[[k]]}, {k, 3, Length[A100094]}]]], n == s^2, 1, IntegerQ[t = n - s^2] && 0 < t < s, Sum[r[s - c - c^2 - t], {c, 0, Floor[-1/2 + (1/2)* Sqrt[1 + 4 s - 4 t]]}], n == s^2 + s, 1, IntegerQ[t = n - s^2 - s] && 0 < t <= s, q[s + 1 - t] + Sum[r[s + 1 - c^2 - t], {c, 1, Floor[Sqrt[s + 1 - t]]}], True, Print["error n = ", n]]];
Select[Table[e[n], {n, 0, nmax}], # <= amax&] (* Jean-François Alcover, Jul 20 2018 *)

It seems that for m >= 1, 0 <= k <= m-1, we have a(m^2-k) = a(k^2+k+1) = A100094(k) and a(m^2+m-k) = a((k+1)^2+1) = A100093(k+1). If this is true, then a(n) = 1 if and only if n is of the form m^2, m^2 + m - 1 or m^2 + m. - Jianing Song, Aug 10 2021

Offset changed to 0 by N. J. A. Sloane, Mar 19 2017

A069813 Maximum number of triangles in polyiamond with perimeter n.

Original entry on oeis.org

1, 2, 3, 6, 7, 10, 13, 16, 19, 24, 27, 32, 37, 42, 47, 54, 59, 66, 73, 80, 87, 96, 103, 112, 121, 130, 139, 150, 159, 170, 181, 192, 203, 216, 227, 240, 253, 266, 279, 294, 307, 322, 337, 352, 367, 384, 399, 416, 433, 450, 467, 486, 503, 522, 541, 560, 579

Offset: 3

Winston C. Yang (winston(AT)cs.wisc.edu), Apr 30 2002

			a(10) = 16 because the maximum number of triangles in a polyiamond of perimeter 10 is 16.

Colin Barker, Table of n, a(n) for n = 3..1000
W. C. Yang, R. R. Meyer, Maximal and minimal polyiamonds, 2002.
Index entries for linear recurrences with constant coefficients, signature (1,1,0,-1,-1,1).

Cf. A000105, A000577, A027709, A030511 (bisection), A057729, A067628.

R:=PowerSeriesRing(Integers(), 65); Coefficients(R!( x^3*(x^2-x-1)*(x^2+1)/((x-1)^3*(x+1)*(x^2+x+1)))); // Marius A. Burtea, Jan 03 2020

A069813 := proc(n)
    round(n^2/6) ;
    if modp(n,6) <> 0 then
        %-1 ;
    else
        % ;
    end if;
end proc: # R. J. Mathar, Jul 14 2015

LinearRecurrence[{1, 1, 0, -1, -1, 1}, {1, 2, 3, 6, 7, 10}, 60] (* Jean-François Alcover, Jan 03 2020 *)

a(n) = round(n^2/6) - (n % 6 != 0) \\ Michel Marcus, Jul 17 2013

Vec(x^3*(x^2-x-1)*(x^2+1)/((x-1)^3*(x+1)*(x^2+x+1)) + O(x^60)) \\ Colin Barker, Jan 19 2015

a(n) = round(n^2/6) - (0 if n = 0 mod 6, 1 else) = A056829(n)-A097325(n).
From Colin Barker, Jan 18 2015: (Start)
a(n) = round((-25 + 9*(-1)^n + 8*exp(-2/3*i*n*Pi) + 8*exp((2*i*n*Pi)/3) + 6*n^2)/36), where i=sqrt(-1).
G.f.: x^3*(1+x-x^2)*(1+x^2) / ((1-x)^3*(1+x)*(1+x+x^2)). (End)
a(n) = A001399(n-3) + A001399(n-4) + A001399(n-6) - A001399(n-7). - R. J. Mathar, Jul 14 2015

cp's OEIS Frontend

A063655 Smallest semiperimeter of integral rectangle with area n.

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A027434 a(1) = 2; then defined by property that a(n) = smallest number >= a(n-1) such that successive runs have lengths 1,1,2,2,3,3,4,4.

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A067628 Minimal perimeter of polyiamond with n triangles.

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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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A083479 The natural numbers with all terms of A033638 inserted.

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A123663 Number of shared edges in a spiral of n unit squares.

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