A231056 The maximum number of X patterns that can be packed into an n X n array of coins.

0, 1, 1, 2, 4, 5, 8, 10, 13, 16, 20, 24, 29, 34, 40, 45, 51, 58, 65, 73, 80, 88, 97, 106, 116, 125, 135, 146, 157, 169, 180, 192, 205, 218, 232, 245, 259, 274, 289, 305, 320, 336, 353, 370, 388, 405, 423, 442, 461, 481, 500, 520, 541, 562, 584, 605, 627, 650, 673, 697, 720, 744, 769, 794

Offset

2,4

Comments

The X pattern (8c5s2 type) is a pattern in which 8 curves cover 5 coins, and is one of a total of 13 such distinct patterns that appear in a tightly-packed 3 X 3 square array of coins of identical size; each of the 8 curves is a circular arc lying along the edge of one of the 5 coins, and the 8 curves are joined end-to-end to form a continuous area.

a(n) is the maximum number of X patterns that can be packed into an n X n array of coins. The total coins left after packing X patterns into an n X n array of coins is A231064 and voids left is A231065.

a(n) is also the maximum number of "+" patterns (8c5s1 type) that can be packed into an n X n array of coins. See illustration in links.

Links

Table of n, a(n) for n=2..65.

Kival Ngaokrajang, Illustration of initial terms (X and +)

Formula

Empirical g.f.: -x^3*(x^15 -2*x^14 +x^13 -x^12 +2*x^11 -2*x^10 +2*x^9 -x^8 +x^5 -x^4 +x^3 +x^2 -x +1) / ((x -1)^3*(x^4 +x^3 +x^2 +x +1)). - Colin Barker, Nov 27 2013

Prog

(Small Basic)
x[2] = 0
d1[3] = 1
For n = 2 To 100
  If Math.Remainder(n+2, 5) = 1 Then
     d2 = 0
  Else
    If Math.Remainder(n+2, 5) = 4 Then
     d2 = -1
    else
     d2 = 1
    EndIf
  EndIf
  d1[n+2] = d1[n+1] + d2
  x[n+1] = x[n] + d1[n+1]
  If n >= 13 And Math.Remainder(n, 5) = 3 Then
    x[n] = x[n] - 1
  EndIf
  If n=6 or n>=16 And Math.Remainder(n, 5)=1 Then
    x[n] = x[n] + 1
  EndIf
  TextWindow.Write(x[n]+", ")
EndFor

Crossrefs

Cf. A008795, A230370 (3-curves); A074148, A227906, A229093, A229154 (4-curves); A001399, A230267, A230276 (5-curves); A229593, A228949, A229598, A002620, A230548, A230549, A230550 (6-curves).

Sequence in context: A144876 A337483 A239517 * A325363 A000549 A191985

Adjacent sequences: A231053 A231054 A231055 * A231057 A231058 A231059

Keyword

nonn

Author

Kival Ngaokrajang, Nov 03 2013

Status

approved