A231065 - OEIS

A231065

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

%I #13 Nov 27 2013 11:25:33

%S 1,0,5,8,9,16,17,24,29,36,41,48,53,60,65,76,85,92,101,108,121,132,141,

%T 152,161,176,189,200,213,224,241,256,269,284,297,316,333,348,365,380,

%U 401,420,437,456,473,496,517,536,557,576,601,624,645,668,689,716,741,764,789,812,841,868

%N Voids left after packing X patterns into an of 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 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.

%C 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.

%H Kival Ngaokrajang, <a href="/A231065/a231065_1.pdf">Illustration of initial terms (V)</a>

%F 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

%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 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 V = (n-1)*(n-1) - x[n]*4

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

%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 2,3

%A _Kival Ngaokrajang_, Nov 03 2013