A085851 -id:A085851 - cp's OEIS frontend

A379043 Decimal expansion of log(A085851).

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

Offset: 0

Stefano Spezia, Dec 14 2024

			0.33324272197618188785374776400567634355947033572296...

Steven R. Finch, Mathematical Constants, Encyclopedia of Mathematics and its Applications, vol. 94, Cambridge University Press, 2003, Section 5.12, p. 343.

Cf. A085851.

RealDigits[Log[Root[-32751691810479015985152 + 97143135277377575190528*#1^4 - 73347491183630103871488*#1^6 - 71220809441400405884928*#1^8 + 107155448150443388043264*#1^10 - 72405670285649161617408*#1^12 + 2958015038376958230528*#1^14 + 7449488310131083100160*#1^16 + 797726698866658379776*#1^18 + 2505062311720673792* #1^20 + 2013290651222784*#1^22 + 25937424601*#1^24 & , 2]], 10,100][[1]] (* after Vaclav Kotesovec, Apr 03 2014 in A085851 *)

A066864 Number of binary arrangements without adjacent 1's on n X n rhombic hexagonal grid.

Original entry on oeis.org

1, 2, 6, 42, 524, 13322, 647252, 61758332, 11435477118, 4129523869606, 2902264461628298, 3973109800760143708, 10590895512774862686570, 54979738656662942307796576, 555797909644630436677137498230, 10941698340065066230952215658836402, 419471520990343359533179780148504998680

Offset: 0

R. H. Hardin, Jan 25 2002

Also the number of tilings of an (n+1) X (n+1) square using 1 X 1 squares and L-tiles. An L-tile is a 2 X 2 square with the upper right 1 X 1 subsquare removed and no rotations are allowed. a(2) = 6:
  .___   ___   ___   ___   ___   ___
  |||_| | ||| |||_| || || |||_| || ||
  |||_| |_|_| | ||| ||__| || || | |_|
  |||_| |||_| |_|_| |||_| ||__| |_|_| - Alois P. Heinz, Jun 06 2013

			Neighbors for n=4:
  o--o--o--o
  | /| /| /|
  |/ |/ |/ |
  o--o--o--o
  | /| /| /|
  |/ |/ |/ |
  o--o--o--o
  | /| /| /|
  |/ |/ |/ |
  o--o--o--o

Steven R. Finch, Mathematical Constants, Cambridge, 2003, pp. 342-349.
J. Katzenelson and R. P. Kurshan, S/R: A Language for Specifying Protocols and Other Coordinating Processes, pp. 286-292 in Proc. IEEE Conf. Comput. Comm., 1986.

Vaclav Kotesovec and Alois P. Heinz, Table of n, a(n) for n = 0..28
Steven R. Finch, Hard Square Entropy Constant [Broken link]
Steven R. Finch, Hard Square Entropy Constant [From the Wayback machine]
Vaclav Kotesovec, Non-attacking chess pieces, 6ed, 2013, pp. 69-71.

Cf. A006506, A027683, A066863, A066865, A066866.

Main diagonal of A219741 and A226444.

a:= proc(n) option remember; local b; b:=
      proc(n, l) option remember; local k;
        if n<2 then 1
      elif min(l[])>0 then b(n-1, map(h->h-1, l))
      else for k while l[k]>0 do od; b(n, subsop(k=1, l))+
           `if`(n>1 and kAlois P. Heinz, Aug 26 2013

$RecursionLimit = 1000; a[n0_] := a[n0] = Module[{b}, b[n_, l_List] := b[n, l] = Module[{k}, Which[n<2, 1, Min[l]>0, b[n-1, l-1], True, For[k = 1, l[[k]] > 0, k++]; b[n, ReplacePart[l, k -> 1]] + If[n>1 && k 2, k+1 -> 1}]], 0]]];  b[n0+1, Array[0&, n0+1]]]; Table[a[n], {n, 0, 15}] (* Jean-François Alcover, Feb 24 2015, after Alois P. Heinz *)

Limit_{n->oo} a(n)^(1/n^2) = 1.395485972... (see A085851).

a(12)-a(21) from Vaclav Kotesovec, May 01 2012
a(0) and a(22) from Alois P. Heinz, Aug 26 2013
a(23) from Alois P. Heinz, Aug 28 2013
a(24) from Vaclav Kotesovec, Sep 19 2014
a(25) from Alois P. Heinz, Dec 03 2014
a(26)-a(28) from Vaclav Kotesovec, Aug 13 2016

A085850 Decimal expansion of hard square entropy constant.

Original entry on oeis.org

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

Offset: 1

Eric W. Weisstein, Jul 05 2003

			1.503048082...

Steven R. Finch, Mathematical Constants, Encyclopedia of Mathematics and its Applications, vol. 94, Cambridge University Press, 2003, Section 5.12, p. 342.

R. J. Baxter, Planar Lattice Gases with Nearest-Neighbour Exclusion, arXiv:cond-mat/9811264 [cond-mat.stat-mech], 1998; Annals of Combinatorics, 3 (1999), pp. 191-203.
S. R. Finch, Several Constants Arising in Statistical Mechanics, Annals Combinat. vol 3 (1999) issue (2-4) pp. 323-335.
Vaclav Kotesovec, Non-attacking chess pieces, 6ed, 2013, p. 69 and 372.
Kai Liang, Independent Set Enumeration and Estimation of Related Constants of Grid Graphs, arXiv:2507.04007 [math.CO], 2025. See p. 8.
Eric Weisstein's World of Mathematics, Hard Square Entropy Constant.

Cf. A006506, A085851, A247413.

More terms from R. J. Mathar, Jul 18 2007

Terms a(19)-a(44) computed R. J. Baxter, 1998

A247413 Decimal expansion of the entropy constant related to A063443.

Original entry on oeis.org

1, 3, 4, 2, 6, 4, 3, 9, 5, 1, 1, 2, 4

Offset: 1

Vaclav Kotesovec, Sep 16 2014

			1.342643951124...

Steven R. Finch, Mathematical Constants, Cambridge, 2003, p. 343.
B. D. McKay, On Calkin and Wilf's limit theorem for grid graphs, unpublished note, 1996.

N. J. Calkin, K. James, and S. Purvis, Counting Kings, 2006, p.9.
N. J. Calkin and H. S. Wilf, The number of independent sets in a grid graph, preprint, 1995.
N. J. Calkin and H. S. Wilf, The number of independent sets in a grid graph, SIAM J. Discrete Math, 11 (1998) 54-60.
Steven R. Finch, Several Constants Arising in Statistical Mechanics, arXiv:math/9810155 [math.CO], 1999, p.8.
Vaclav Kotesovec, Non-attacking chess pieces, 6ed, 2013, p.68-69.
D. E. Knuth, Nonattacking kings on a chessboard, 1994.
M. Larsen, The Problem of Kings, The Electronic Journal of Combinatorics 2, 1995.
Kai Liang, Independent Set Enumeration in King Graphs by Tensor Network Contractions, arXiv:2505.12776 [math.CO], 2025. See p. 6.
Eric Weisstein's World of Mathematics, Hard Hexagon Entropy Constant
Eric Weisstein's World of Mathematics, Hard Square Entropy Constant
H. S. Wilf, The problem of the kings, Elec. J. Combin. 2, 1995.

Cf. A063443, A212269.

Cf. A085850, A085851.

Equals limit n->infinity (A063443(n))^(1/n^2).

Equals limit n->infinity (A212269(n))^(1/n^2).

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A379043 Decimal expansion of log(A085851).

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A066864 Number of binary arrangements without adjacent 1's on n X n rhombic hexagonal grid.

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A085850 Decimal expansion of hard square entropy constant.

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A247413 Decimal expansion of the entropy constant related to A063443.

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