A261491 -id:A261491 - cp's OEIS frontend

A001971 Nearest integer to n^2/8.

0, 0, 1, 1, 2, 3, 5, 6, 8, 10, 13, 15, 18, 21, 25, 28, 32, 36, 41, 45, 50, 55, 61, 66, 72, 78, 85, 91, 98, 105, 113, 120, 128, 136, 145, 153, 162, 171, 181, 190, 200, 210, 221, 231, 242, 253, 265, 276, 288, 300, 313, 325, 338, 351, 365, 378, 392, 406, 421, 435, 450

Offset: 0

N. J. A. Sloane

Restricted partitions.
a(0) = a(1) = 0; a(n) are the partitions of floor((3*n+3)/2) with 3 distinct numbers of the set {1, ..., n}; partitions of floor((3*n+3)/2)-C and ceiling((3*n+3)/2)+C have equal numbers. - Paul Weisenhorn, Jun 05 2009, corrected by M. F. Hasler, Jun 16 2022
Odd-indexed terms are the triangular numbers, even-indexed terms are the midpoint (rounded up where necessary) of the surrounding odd-indexed terms. - Carl R. White, Aug 12 2010
a(n+2) is the number of points one can surround with n stones in Go (including the points under the stones). - Thomas Dybdahl Ahle, May 11 2014
Corollary of above: a(n) is the number of points one can surround with n+2 stones in Go (excluding the points under the stones). - Juhani Heino, Aug 29 2015
From Washington Bomfim, Jan 13 2021: (Start)
For n >= 4, a(n) = A026810(n+2) - A026810(n-4).
Let \n,m\ be the number of partitions of n into m non-distinct parts.
For n >= 1, \n,4\ = round((n-2)^2/8).
For n >= 6, \n,4\ = A026810(n) - A026810(n-6).
(End)

A. Cayley, Numerical tables supplementary to second memoir on quantics, Collected Mathematical Papers. Vols. 1-13, Cambridge Univ. Press, London, 1889-1897, Vol. 2, pp. 276-281.
M. Jeger, Einfuehrung in die Kombinatorik, Klett, 1975, Bd.2, pages 110 ff. [Paul Weisenhorn, Jun 05 2009]
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Vincenzo Librandi, Table of n, a(n) for n = 0..10000
G. Almkvist, Invariants, mostly old ones, Pacific J. Math. 86 (1980), no. 1, 1-13. MR0586866 (81j:14029)
A. Cayley, Numerical tables supplementary to second memoir on quantics, Collected Mathematical Papers. Vols. 1-13, Cambridge Univ. Press, London, 1889-1897, Vol. 2, pp. 276-281. [Annotated scanned copy]
Shalosh B. Ekhad and Doron Zeilberger, In How many ways can I carry a total of n coins in my two pockets, and have the same amount in both pockets?, arXiv:1901.08172 [math.CO], 2019.
Simon Plouffe, Approximations de séries génératrices et quelques conjectures, Dissertation, Université du Québec à Montréal, 1992; arXiv:0911.4975 [math.NT], 2009.
Simon Plouffe, 1031 Generating Functions, Appendix to Thesis, Montreal, 1992.
D. Vainsencher and A. M. Bruckstein, On isoperimetrically optimal polyforms, Theoretical Computer Science 406.1-2, 2008, pp. 146-159.
Index entries for linear recurrences with constant coefficients, signature (2,-1,0,1,-2,1).

The 4th diagonal of A061857?
Kind of an inverse of A261491 (regarding Go).
Cf. A026810 (partitions with greatest part 4), A001400 (partitions in at most 4 parts), A000217 (a(2n+1): triangular numbers n(n+1)/2), A000982 (a(2n): round(n^2/2)).

a001971 = floor . (+ 0.5) . (/ 8) . fromIntegral . (^ 2)
-- Reinhard Zumkeller, May 08 2012

[Round(n^2/8): n in [0..60]]; // Vincenzo Librandi, Jun 23 2011

A001971:=-(1-z+z**2)/((z+1)*(z**2+1)*(z-1)**3); # conjectured (correctly) by Simon Plouffe in his 1992 dissertation [Note that this "generating function" is Sum_{n >= 0} a(n+2)*z^n, not a(n)*z^n. - M. F. Hasler, Jun 16 2022]

LinearRecurrence[{2,-1,0,1,-2,1},{0,0,1,1,2,3},70] (* Harvey P. Dale, Jan 30 2014 *)

PARI
```
{a(n) = round(n^2 / 8)};
```

apply( {A001971(n)=n^2\/8}, [0..99]) \\ M. F. Hasler, Jun 16 2022

The listed terms through a(20)=50 satisfy a(n+2) = a(n-2) + n. - John W. Layman, Dec 16 1999
G.f.: x^2 * (1 - x + x^2) / (1 - 2*x + x^2 - x^4 + 2*x^5 - x^6) = x^2 * (1 - x^6) / ((1 - x) * (1 - x^2) * (1 - x^3) * (1 - x^4)). - Michael Somos, Feb 07 2004
a(n) = floor((n^2+4)/8). - Paul Weisenhorn, Jun 05 2009
a(2*n+1) = A000217(n), a(2*n) = floor((A000217(n-1)+A000217(n)+1)/2). - Carl R. White, Aug 12 2010
From Michael Somos, Aug 29 2015: (Start)
Euler transform of length 6 sequence [ 1, 1, 1, 1, 0, -1].
a(n) = a(-n) for all n in Z. (End)
a(2n) = A000982(n). - M. F. Hasler, Jun 16 2022
Sum_{n>=2} 1/a(n) = 2 + Pi^2/12 + tanh(Pi/2)*Pi/2. - Amiram Eldar, Jul 02 2023

Edited Feb 08 2004

A275113 a(n) is the minimal number of squares needed to enclose n squares with a wall so that there is a gap of at least one cell between the wall and the enclosed cells.

Original entry on oeis.org

12, 14, 15, 16, 16, 17, 18, 18, 19, 19, 20, 20, 20, 21, 21, 22

Offset: 1

Kival Ngaokrajang, Jul 17 2016

Inspired by beehive construction in which wax is used in the most efficient way. This problem is likened to construction of a fence around a house with minimum materials and maximum enclosed area. I conjectured that a specific house pattern shall be selected. See illustration in links.
If the conjecture in A261491 is true (i.e., A261491(n) is the number of squares required to enclose n squares without a gap), then a(n) = A261491(n) + 8. - Charlie Neder, Jul 11 2018
The conjecture in A261491 holds through a(16). - David Consiglio, Jr., Nov 10 2022

			     a(1) = 12:
     +--+--+--+
     | 1| 2| 3|
  +--+--+--+--+--+
  |12|        | 4|
  +--+  +--+  +--+
  |11|  | 1|  | 5|
  +--+  +--+  +--+
  |10|        | 6|
  +--+--+--+--+--+
     | 9| 8| 7|
     +--+--+--+
.
      a(2) = 14:
     +--+--+--+--+
     | 1| 2| 3| 4|
  +--+--+--+--+--+--+
  |14|           | 5|
  +--+  +--+--+  +--+
  |13|  | 1| 2|  | 6|
  +--+  +--+--+  +--+
  |12|           | 7|
  +--+--+--+--+--+--+
     |11|10| 9| 8|
     +--+--+--+--+
.
     a(3) = 15:
     +--+--+--+
     | 1| 2| 3|
  +--+--+--+--+--+
  |15|        | 4|
  +--+  +--+  +--+--+
  |14|  | 3|     | 5|
  +--+  +--+--+  +--+
  |13|  | 1| 2|  | 6|
  +--+  +--+--+  +--+
  |12|           | 7|
  +--+--+--+--+--+--+
     |11|10| 9| 8|
     +--+--+--+--+

David Consiglio, Jr., Python Program
David Consiglio, Jr., Illustration of a(11)
David Consiglio, Jr., Illustration of a(12)
David Consiglio, Jr., Illustration of a(13)
David Consiglio, Jr., Illustration of a(14)
David Consiglio, Jr., Illustration of a(15)
David Consiglio, Jr., Illustration of a(16)
Kival Ngaokrajang, Illustration of initial terms

Cf. A235382, A261491.

a(11)-a(16) from David Consiglio, Jr., Nov 10 2022

A275937 The number of distinct patterns of the smallest number of unit squares required to enclose n units of area, where corner contact is allowed.

Original entry on oeis.org

1, 1, 2, 1, 3, 1, 1, 4, 2, 4, 1

Offset: 0

Kival Ngaokrajang, Aug 12 2016

Inspired by A235382 and A261491. The rotations and/or reflections are excluded.

Kival Ngaokrajang, Illustration of initial terms

Cf. A235382, A261491, A100092.

A346958 a(n) is the minimal number of cubes required to make a void of volume n.

Original entry on oeis.org

6, 10, 13, 15, 17, 18, 18, 21, 23, 25, 26, 26

Offset: 1

Mohammed Yaseen, Aug 08 2021

Following is an illustration of the first few voids in the form of polycubes (where an o represents a continuation upwards and an x represents a continuation downwards) each of which can be made by concealing it with a(n) cubes.
                          .---.       .---.
                          |   |       |   |
  .---.   .---.---.   .---.---.   .---.---.
  |   |   |   |   |   |   |   |   |   | o |
  .---.   .---.---.   .---.---.   .---.---.
   n=1       n=2         n=3         n=4
      .---.       .---.           .---.
      |   |       |   |           |   |
  .---.---.   .---.---.---.   .---.---.---.
  |   | o |   |   | o |   |   |   | ox|   |
  .---.---.   .---.---.---.   .---.---.---.
      |   |       |   |           |   |
      .---.       .---.           .---.
     n=5           n=6             n=7
Equivalently, the minimum perimeter size of any polycube of size n. - Sean A. Irvine, Aug 23 2021
Conjecture: When n is in A001845 the void is an octahedral crystal ball of volume n = A001845(m), which is concealed by a(n) = A005899(m+1) cubes. So a(A001845(m)) = A005899(m+1), m>=0. For example, a(1)=6 and a(7)=18. - Mohammed Yaseen, Sep 15 2022

			A cube-shaped void can be made by concealing it with 6 cubes, which is the minimal number to do so. So a(1)=6.
A dicube-shaped void can be made by concealing it with 10 cubes, which is the minimal number to do so. So a(2)=10.

Sean A. Irvine, Java program (github)

Cf. A261491 (2D analog).

Cf. A000162, A003211, A193416.

a(n) < A193416(n) for n>2.

a(8)-a(12) from Sean A. Irvine, Aug 23 2021

cp's OEIS Frontend

A001971 Nearest integer to n^2/8.

Views

Author

Keywords

Comments

References

Links

Crossrefs

Programs

Haskell

Magma

Maple

Mathematica

PARI

PARI

Formula

Extensions

A275113 a(n) is the minimal number of squares needed to enclose n squares with a wall so that there is a gap of at least one cell between the wall and the enclosed cells.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Extensions

A275937 The number of distinct patterns of the smallest number of unit squares required to enclose n units of area, where corner contact is allowed.

Views

Author

Keywords

Comments

Links

Crossrefs

A346958 a(n) is the minimal number of cubes required to make a void of volume n.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Formula

Extensions