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A230276 Voids left after packing 5-curves coins patterns into fountain of coins with base n.

%I A230276 #22 May 05 2023 12:22:57
%S A230276 0,1,1,6,10,16,24,34,43,57,70,85,102,121,139,162,184,208,234,262,289,
%T A230276 321,352,385,420,457,493,534,574,616,660,706,751,801,850,901,954,1009,
%U A230276 1063,1122,1180,1240,1302,1366,1429
%N A230276 Voids left after packing 5-curves coins patterns into fountain of coins with base n.
%C A230276 Refer to arrangement same as A005169: "A fountain is formed by starting with a row of coins, then stacking additional coins on top so that each new coin touches two in the previous row". The 5 curves coins patterns consist of a part of each coin circumference and forms a continuous area. There are total 13 distinct patterns. For selected pattern, I would like to call "5C4S" type as it cover 4 coins and symmetry. When packing 5C4S into fountain of coins base n, the total number of 5C4S is A001399, the coins left is A230267 and void left is a(n). See illustration in links.
%H A230276 Kival Ngaokrajang, <a href="/A230276/a230276.pdf">Illustration of initial terms (V)</a>
%H A230276 <a href="/index/Rec#order_06">Index entries for linear recurrences with constant coefficients</a>, signature (1,1,0,-1,-1,1).
%F A230276 G.f.: x^2*(x^4 + 3*x^3 + 4*x^2 + 1)/((1-x)*(1-x^2)*(1-x^3)). - _Ralf Stephan_, Oct 17 2013
%F A230276 a(n) = (9*(-1)^n+18*n^2-48*n)/24 - A099837(n)/3. - _R. J. Mathar_, Feb 28 2018
%p A230276 A099837 := proc(n)
%p A230276     op(modp(n,3)+1,[2,-1,-1]) ;
%p A230276 end proc:
%p A230276 A230276 := proc(n)
%p A230276     -A099837(n)/3 + (-48*n+31+18*n^2+9*(-1)^n)/24 ;
%p A230276 end proc:
%p A230276 seq(A230276(n),n=1..40) ; # _R. J. Mathar_, Feb 28 2018
%t A230276 LinearRecurrence[{1, 1, 0, -1, -1, 1}, {0, 1, 1, 6, 10, 16}, 45] (* _Jean-François Alcover_, May 05 2023 *)
%o A230276 (Small Basic)
%o A230276 a[1]=0
%o A230276 d[2]=1
%o A230276 For n = 1 To 100
%o A230276   If n+1 >= 3 Then
%o A230276     If Math.Remainder(n+1,3)=math.Remainder(n+1,6) Then
%o A230276       d2=2
%o A230276     Else
%o A230276      If Math.Remainder(n+1,3)+math.Remainder(n+1,6)=5 then
%o A230276        d2=5
%o A230276      Else
%o A230276        d2=-1
%o A230276      EndIf
%o A230276     EndIf
%o A230276     d[n+1]=d[n]+d2
%o A230276   EndIf
%o A230276   a[n+1]=a[n]+d[n+1]
%o A230276 TextWindow.Write(a[n]+", ")
%o A230276 EndFor
%Y A230276 Cf. A008795 (3-curves coins patterns), A074148, A229093, A229154 (4-curves coins patterns), A001399 (5-curves coins patterns), A229593 (6-curves coins patterns).
%K A230276 nonn,easy
%O A230276 1,4
%A A230276 _Kival Ngaokrajang_, Oct 15 2013