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A231064 Coins left after packing X patterns into an n X n array of coins. 3

%I

%S 4,4,11,15,16,24,24,31,35,41,44,49,51,55,56,64,69,71,75,76,84,89,91,

%T 95,96,104,109,111,115,116,124,129,131,135,136,144,149,151,155,156,

%U 164,169,171,175,176,184,189,191,195,196,204,209,211,215,216,224,229,231,235,236,244,249,251

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

%C 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 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 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 Kival Ngaokrajang, <a href="/A231064/a231064_1.pdf">Illustration of initial terms (U)</a>

%F 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 (Small Basic)

%o x[2] = 0

%o d1[3] = 1

%o For n = 2 To 100

%o If Math.Remainder(n+2,5) = 1 Then

%o d2 = 0

%o Else

%o If Math.Remainder(n+2,5) = 4 Then

%o d2 = -1

%o else

%o d2 = 1

%o EndIf

%o EndIf

%o d1[n+2] = d1[n+1] + d2

%o x[n+1] = x[n] + d1[n+1]

%o If n >= 13 And Math.Remainder(n,5) = 3 Then

%o x[n] = x[n] - 1

%o EndIf

%o If n=6 or n>=16 And Math.Remainder(n,5)=1 Then

%o x[n] = x[n] + 1

%o EndIf

%o U = n*n - x[n]*5

%o TextWindow.Write(U+", ")

%o EndFor

%Y 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 nonn

%O 2,1

%A _Kival Ngaokrajang_, Nov 03 2013

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Last modified July 9 13:27 EDT 2020. Contains 335543 sequences. (Running on oeis4.)