A231065 Voids left after packing X patterns into an of n X n array of coins.

1, 0, 5, 8, 9, 16, 17, 24, 29, 36, 41, 48, 53, 60, 65, 76, 85, 92, 101, 108, 121, 132, 141, 152, 161, 176, 189, 200, 213, 224, 241, 256, 269, 284, 297, 316, 333, 348, 365, 380, 401, 420, 437, 456, 473, 496, 517, 536, 557, 576, 601, 624, 645, 668, 689, 716, 741, 764, 789, 812, 841, 868

Offset

2,3

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 total number of voids (spaces among coins) left after packing X patterns into an n X n array of coins. The maximum number of X patterns that can be packed into an n X n array of coins is A231056 and coins left is A231064.

a(n) is also the total number of voids left after packing "+" patterns (8c5s1 type) into an n X n array of coins. See illustration in links.

Links

Table of n, a(n) for n=2..63.

Kival Ngaokrajang, Illustration of initial terms (V)

Formula

Empirical g.f.: x^2*(4*x^16 -8*x^15 +4*x^14 -4*x^13 +8*x^12 -8*x^11 +8*x^10 -4*x^9 +4*x^6 -5*x^5 +2*x^4 +2*x^3 -6*x^2 +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
  V = (n-1)*(n-1) - x[n]*4
  TextWindow.Write(V+", ")
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: A334919 A047616 A287551 * A314576 A045221 A046287

Adjacent sequences: A231062 A231063 A231064 * A231066 A231067 A231068

Keyword

nonn

Author

Kival Ngaokrajang, Nov 03 2013

Status

approved