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

0, 1, 1, 6, 10, 16, 24, 34, 43, 57, 70, 85, 102, 121, 139, 162, 184, 208, 234, 262, 289, 321, 352, 385, 420, 457, 493, 534, 574, 616, 660, 706, 751, 801, 850, 901, 954, 1009, 1063, 1122, 1180, 1240, 1302, 1366, 1429

Offset

1,4

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 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.

Links

Table of n, a(n) for n=1..45.

Kival Ngaokrajang, Illustration of initial terms (V)

Index entries for linear recurrences with constant coefficients, signature (1,1,0,-1,-1,1).

Formula

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

a(n) = (9*(-1)^n+18*n^2-48*n)/24 - A099837(n)/3. - R. J. Mathar, Feb 28 2018

Maple

A099837 := proc(n)
    op(modp(n, 3)+1, [2, -1, -1]) ;
end proc:
A230276 := proc(n)
    -A099837(n)/3 + (-48*n+31+18*n^2+9*(-1)^n)/24 ;
end proc:
seq(A230276(n), n=1..40) ; # R. J. Mathar, Feb 28 2018

Mathematica

LinearRecurrence[{1, 1, 0, -1, -1, 1}, {0, 1, 1, 6, 10, 16}, 45] (* Jean-François Alcover, May 05 2023 *)

Prog

(Small Basic)
a[1]=0
d[2]=1
For n = 1 To 100
  If n+1 >= 3 Then
    If Math.Remainder(n+1, 3)=math.Remainder(n+1, 6) Then
      d2=2
    Else
     If Math.Remainder(n+1, 3)+math.Remainder(n+1, 6)=5 then
       d2=5
     Else
       d2=-1
     EndIf
    EndIf
    d[n+1]=d[n]+d2
  EndIf
  a[n+1]=a[n]+d[n+1]
TextWindow.Write(a[n]+", ")
EndFor

Crossrefs

Cf. A008795 (3-curves coins patterns), A074148, A229093, A229154 (4-curves coins patterns), A001399 (5-curves coins patterns), A229593 (6-curves coins patterns).

Sequence in context: A315350 A340773 A114975 * A079329 A302748 A020741

Adjacent sequences: A230273 A230274 A230275 * A230277 A230278 A230279

Keyword

nonn,easy

Author

Kival Ngaokrajang, Oct 15 2013

Status

approved