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A231064 Coins left after packing X patterns into an n X n array of coins.

%I A231064 #14 Nov 27 2013 11:25:54
%S A231064 4,4,11,15,16,24,24,31,35,41,44,49,51,55,56,64,69,71,75,76,84,89,91,
%T A231064 95,96,104,109,111,115,116,124,129,131,135,136,144,149,151,155,156,
%U A231064 164,169,171,175,176,184,189,191,195,196,204,209,211,215,216,224,229,231,235,236,244,249,251
%N A231064 Coins left after packing X patterns into an n X n array of coins.
%C A231064 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 A231064 a(n) is the total number of coins left (the coins out side X patterns) 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 voids left is A231065.
%C A231064 a(n) is also the total number of coins left after packing "+" patterns (8c5s1 type) into an n X n array of coins. See illustration in links.
%H A231064 Kival Ngaokrajang, <a href="/A231064/a231064_1.pdf">Illustration of initial terms (U)</a>
%F A231064 Empirical g.f.: x^2*(5*x^15 -5*x^14 -5*x^12 +5*x^11 -5*x^10 +5*x^9 +4*x^5 +x^4 +4*x^3 +7*x^2 +4) / ((x -1)^2*(x^4 +x^3 +x^2 +x +1)). - _Colin Barker_, Nov 27 2013
%o A231064 (Small Basic)
%o A231064 x[2] = 0
%o A231064 d1[3] = 1
%o A231064 For n = 2 To 100
%o A231064   If Math.Remainder(n+2,5) = 1 Then
%o A231064      d2 = 0
%o A231064   Else
%o A231064     If Math.Remainder(n+2,5) = 4 Then
%o A231064      d2 = -1
%o A231064     else
%o A231064      d2 = 1
%o A231064     EndIf
%o A231064   EndIf
%o A231064   d1[n+2] = d1[n+1] + d2
%o A231064   x[n+1] = x[n] + d1[n+1]
%o A231064   If n >= 13 And Math.Remainder(n,5) = 3 Then
%o A231064     x[n] = x[n] - 1
%o A231064   EndIf
%o A231064   If n=6 or n>=16 And Math.Remainder(n,5)=1 Then
%o A231064     x[n] = x[n] + 1
%o A231064   EndIf
%o A231064   U = n*n - x[n]*5
%o A231064   TextWindow.Write(U+", ")
%o A231064 EndFor
%Y A231064 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 A231064 nonn
%O A231064 2,1
%A A231064 _Kival Ngaokrajang_, Nov 03 2013