A230370 Voids left after packing 3 curves coins patterns (3c3s type) into fountain of coins base n.

0, 0, 3, 6, 13, 19, 39, 54, 66, 85, 100, 123, 141, 168, 189, 220, 244, 279, 306, 345, 375, 418, 451, 498, 534, 585, 624, 679, 721, 780, 825, 888, 936, 1003, 1054, 1125, 1179, 1254, 1311, 1390, 1450, 1533, 1596, 1683

Offset

1,3

Comments

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 3 curves coins patterns consist of a part of each coin circumference and forms a continuous area. There are total 4 distinct patterns. For selected pattern, I would like to call "3c3s" type as it cover 3 coins and symmetry. When packing 3c3s into fountain of coins base n, the total number of 3c3s is A008805, the coins left is A008795 and voids left is a(n). See illustration in links.

Links

Table of n, a(n) for n=1..44.

Kival Ngaokrajang, Illustration of initial terms (V)

Formula

G.f.: x^3*(11*x^8 - 5*x^7 - 21*x^6 + 6*x^5 + 9*x^4 + x^2 + 3*x + 3)/((1-x)*(1-x^2)^2) (conjectured). Ralf Stephan, Oct 19 2013

Prog

(Small Basic)
a[1]=0
a[2]=0
d1[3]=3
For n=1 To 100
  If n+2>=4 Then
    If Math.Remainder(n+2, 2)=0 Then
      d2= 2-(n+2)/2
    Else
      d2= (n+5)/2
    EndIf
      d1[n+2]=d1[n+1]+d2
  EndIf
  a[n+2]=a[n+1]+d1[n+2]
  TextWindow.Write(a[n]+", ")
EndFor

Crossrefs

A001399, A230267, A230276 (5-curves coins patterns); A074148, A229093, A220154 (4-curves coins patterns); A008795 (3-curves coins patterns).

Sequence in context: A259583 A064349 A101965 * A285246 A147009 A361273

Adjacent sequences: A230367 A230368 A230369 * A230371 A230372 A230373

Keyword

nonn

Author

Kival Ngaokrajang, Oct 17 2013

Status

approved