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A227906 Coins left after packing heart patterns (fixed orientation) into n X n coins.

%I A227906 #31 Jan 31 2022 19:41:00
%S A227906 2,4,4,9,6,13,8,17,10,21,12,25,14,29,16,33,18,37,20,41,22,45,24,49,26,
%T A227906 53,28,57,30,61,32,65,34,69,36,73,38,77,40,81,42,85,44,89,46,93,48,97,
%U A227906 50,101,52,105,54,109,56,113,58,117,60,121
%N A227906 Coins left after packing heart patterns (fixed orientation) into n X n coins.
%C A227906 On the Japanese TV show "Tsuki no Koibito", a girl told her boyfriend that she saw a heart in 4 coins. Actually there are a total of 6 distinct patterns appearing in 2 X 2 coins in which each pattern consists of a part of the perimeter of each coin and forms a continuous area.
%C A227906 a(n) is the number of coins left after packing fixed orientation heart patterns (type 4c2s1: 4-curve cover 2 coins and symmetry) into n X n coins. The total number of hearts is A093005 and the number of voids left is A093353. See illustration in links.
%H A227906 Kival Ngaokrajang, <a href="/A227906/a227906.pdf">Illustration of initial terms (U)</a>
%H A227906 <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (0,2,0,-1).
%F A227906 From _Colin Barker_, Oct 30 2013: (Start)
%F A227906 a(n) = (-1 + (-1)^n - (-3 + (-1)^n)*n)/2 for n>3.
%F A227906 a(n) = n for n>3 and even.
%F A227906 a(n) = 2*n-1 for n > 3 and odd.
%F A227906 a(n) = 2*a(n-2) - a(n-4) for n>7.
%F A227906 G.f.: -x^2*(x^5-x^3-4*x-2) / ((x-1)^2*(x+1)^2).(End)
%t A227906 With[{nn=60},Join[{2,4},Riffle[Range[4,nn,2],Range[9,2nn+1,4]]]] (* _Harvey P. Dale_, Feb 11 2015 *)
%o A227906 (Small Basic)
%o A227906 For n = 2 To 100
%o A227906   If Math.Remainder(n,2) = 0 then
%o A227906     a = n
%o A227906   Else
%o A227906     a = a + n
%o A227906     If n = 3 then
%o A227906       a = a - 1
%o A227906     endif
%o A227906   EndIf
%o A227906   TextWindow.Write(a+", ")
%o A227906 EndFor
%o A227906 (PARI) Vec(-x^2*(x^5-x^3-4*x-2)/((x-1)^2*(x+1)^2) + O(x^100)) \\ _Colin Barker_, Oct 30 2013
%Y A227906 Cf. A008795, A230370 (3-curves), A074148, A229093, A229154 (4-curves), A001399, A230267, A230276 (5-curves), A229593, A228949, A229598 (6-curves).
%K A227906 nonn,easy
%O A227906 2,1
%A A227906 _Kival Ngaokrajang_, Oct 19 2013