

A231056


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


3



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
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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 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 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



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


KEYWORD

nonn


AUTHOR



STATUS

approved



