

A231065


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


3



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
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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 tightlypacked 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 endtoend 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 = (n1)*(n1)  x[n]*4
TextWindow.Write(V+", ")
EndFor


CROSSREFS

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



