A230549 - OEIS

%I #16 Dec 18 2015 18:18:44

%S 4,5,8,13,12,21,32,33,40,57,48,69,84,85,96,125,108,141,160,161,176,

%T 217,192,237,260,261,280,333,300,357,384,385,408,473,432,501,532,533,

%U 560,637,588,669,704,705,736,825,768,861

%N Coins left after packing twin hearts patterns into n X n coins.

%C Twin hearts (6c4a type) is one of total 17 distinct patterns appearing in 3X2 coins where each pattern consists of 6 perimeter parts from each coin and forms a continuous area.

%C a(n) is total coins left after packing twin hearts patterns (6c4a type: 6-curves cover 4 coins) into n X n coins with rotation not allowed. The total twin hearts patterns is A230548 and voids left is A230550. See illustration in links.

%H Kival Ngaokrajang, <a href="/A230549/a230549.pdf">Illustration of initial terms (U)</a>

%F a(n) = n^2 - 4*A230548(n).

%F G.f.: x^2 * (-3*x^10 - 4*x^8 + 3*x^7 + 8*x^6 + 4*x^5 - x^4 + 4*x^3 + 4*x^2 + 5*x + 4)/(1+x^3)*(1-x^3)^2*(1-x^2). (conjectured). - _Ralf Stephan_, Oct 30 2013

%o (Small Basic)

%o  col = 1

%o row = 0

%o For n = 2 To 100

%o   add = 0

%o   If Math.Remainder(n,2) * Math.Remainder(n,3) <> 0 Then

%o     add = 1

%o   EndIf

%o   If n >= 4 And Math.Remainder(n,2) = 0 Then

%o     col = col + 1

%o   EndIf

%o   If n >= 3 And Math.Remainder(n,3) = 0 Then

%o     row = row + 1

%o   EndIf

%o   U = n * n - (col * row + add)*4

%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 (6-curves).

%K nonn

%O 2,1

%A _Kival Ngaokrajang_, Oct 23 2013