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

%I A231056 #16 Nov 27 2013 11:26:38
%S A231056 0,1,1,2,4,5,8,10,13,16,20,24,29,34,40,45,51,58,65,73,80,88,97,106,
%T A231056 116,125,135,146,157,169,180,192,205,218,232,245,259,274,289,305,320,
%U A231056 336,353,370,388,405,423,442,461,481,500,520,541,562,584,605,627,650,673,697,720,744,769,794
%N A231056 The maximum number of X patterns that can be packed into an n X n array of coins.
%C A231056 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.
%C A231056 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.
%C A231056 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.
%H A231056 Kival Ngaokrajang, <a href="/A231056/a231056_1.pdf">Illustration of initial terms (X and +)</a>
%F A231056 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
%o A231056 (Small Basic)
%o A231056 x[2] = 0
%o A231056 d1[3] = 1
%o A231056 For n = 2 To 100
%o A231056   If Math.Remainder(n+2,5) = 1 Then
%o A231056      d2 = 0
%o A231056   Else
%o A231056     If Math.Remainder(n+2,5) = 4 Then
%o A231056      d2 = -1
%o A231056     else
%o A231056      d2 = 1
%o A231056     EndIf
%o A231056   EndIf
%o A231056   d1[n+2] = d1[n+1] + d2
%o A231056   x[n+1] = x[n] + d1[n+1]
%o A231056   If n >= 13 And Math.Remainder(n,5) = 3 Then
%o A231056     x[n] = x[n] - 1
%o A231056   EndIf
%o A231056   If n=6 or n>=16 And Math.Remainder(n,5)=1 Then
%o A231056     x[n] = x[n] + 1
%o A231056   EndIf
%o A231056   TextWindow.Write(x[n]+", ")
%o A231056 EndFor
%Y A231056 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).
%K A231056 nonn
%O A231056 2,4
%A A231056 _Kival Ngaokrajang_, Nov 03 2013