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A229593 Number of boomerang patterns appearing in n X n coins, rotation not allowed.

%I A229593 #46 Sep 08 2022 08:46:06
%S A229593 0,2,3,4,10,12,14,24,27,30,44,48,52,70,75,80,102,108,114,140,147,154,
%T A229593 184,192,200,234,243,252,290,300,310,352,363,374,420,432,444,494,507,
%U A229593 520,574,588,602,660,675,690,752,768
%N A229593 Number of boomerang patterns appearing in n X n coins, rotation not allowed.
%C A229593 The boomerang pattern is one of a total of 17 distinct patterns appearing in a 3 X 2 rectangular array of coins where each pattern consists of perimeter parts from each of 6 coins and forms a continuous area. See illustration of 6-curve patterns in links.
%C A229593 a(n) is the number of boomerang patterns appearing in an n X n array of coins with rotation not allowed. The number of inverse patterns is given in A229598.
%C A229593 It appears that a(n+1) is equivalent to n multiplied by the least possible number of addends in the partition in which the addends are multiplied together to produce the largest possible product for all n > 2. E.g., in the case of a(11), we look for partitions of 10, and for each partition we take the product of all its addends. The largest possible product formed is 3*3*2*2 = 3*3*4 = 36. The least possible number of addends here is 3, which we multiply by 10 to get 30. - _Laurance L. Y. Lau_, Jun 22 2015
%H A229593 Vincenzo Librandi, <a href="/A229593/b229593.txt">Table of n, a(n) for n = 2..1000</a>
%H A229593 Kival Ngaokrajang, <a href="/A229593/a229593.pdf">Illustration of initial terms</a>
%H A229593 Kival Ngaokrajang, <a href="/A229593/a229593_2.pdf">Illustration of 6-curve patterns</a>
%H A229593 <a href="/index/Rec#order_07">Index entries for linear recurrences with constant coefficients</a>, signature (1,0,2,-2,0,-1,1).
%F A229593 G.f.: (2*x^6 + x^5 + x^4 + 2*x^3)/((1-x^3)^2 * (1-x)). - _Ralf Stephan_, Oct 05 2013
%F A229593 3*a(n) = (1-n)^2 -2*A057078(n) +(-1)^n*A110665(n+1). - _R. J. Mathar_, Oct 09 2013
%F A229593 a(n) = (n-1)*floor(n/3). - _Laurance L. Y. Lau_, Jun 22 2015
%t A229593 CoefficientList[Series[(2 x^4 + x^3 + x^2 + 2 x)/((1 - x^3)^2 (1 - x)), {x, 0, 80}], x] (* _Vincenzo Librandi_, Oct 10 2013 *)
%o A229593 (Small Basic)
%o A229593 b[2]=0
%o A229593 d[3]=2
%o A229593 d[4]=1
%o A229593 d[5]=1
%o A229593 For n=2 To 100
%o A229593   If n+1 >=6 Then
%o A229593     If Math.Remainder(n+1,3)=0 Then
%o A229593       d[n+1]=d[n-2]+4
%o A229593     Else
%o A229593       d[n+1]=d[n-2]+1
%o A229593     EndIf
%o A229593   EndIf
%o A229593   b[n+1]=b[n]+d[n+1]
%o A229593   TextWindow.Write(b[n]+", ")
%o A229593 EndFor
%o A229593 (PARI) a(n)=([0,1,0,0,0,0,0; 0,0,1,0,0,0,0; 0,0,0,1,0,0,0; 0,0,0,0,1,0,0; 0,0,0,0,0,1,0; 0,0,0,0,0,0,1; 1,-1,0,-2,2,0,1]^(n-2)*[0;2;3;4;10;12;14])[1,1] \\ _Charles R Greathouse IV_, Jun 16 2015
%o A229593 (Magma) [(n-1)*Floor(n/3): n in [2..60]]; // _Vincenzo Librandi_, Jul 09 2015
%Y A229593 Cf. A074148 (Heart patterns), A229093 (Clubs patterns - fixed orientation), A229154 (Clubs Patterns - rotation allowed)
%K A229593 nonn,easy
%O A229593 2,2
%A A229593 _Kival Ngaokrajang_, Sep 26 2013
%E A229593 G.f. adapted to the offset by _Vincenzo Librandi_, Oct 10 2013