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

0, 2, 3, 4, 10, 12, 14, 24, 27, 30, 44, 48, 52, 70, 75, 80, 102, 108, 114, 140, 147, 154, 184, 192, 200, 234, 243, 252, 290, 300, 310, 352, 363, 374, 420, 432, 444, 494, 507, 520, 574, 588, 602, 660, 675, 690, 752, 768

Offset

2,2

Comments

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.

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.

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

Links

Vincenzo Librandi, Table of n, a(n) for n = 2..1000

Kival Ngaokrajang, Illustration of initial terms

Kival Ngaokrajang, Illustration of 6-curve patterns

Index entries for linear recurrences with constant coefficients, signature (1,0,2,-2,0,-1,1).

Formula

G.f.: (2*x^6 + x^5 + x^4 + 2*x^3)/((1-x^3)^2 * (1-x)). - Ralf Stephan, Oct 05 2013

3*a(n) = (1-n)^2 -2*A057078(n) +(-1)^n*A110665(n+1). - R. J. Mathar, Oct 09 2013

a(n) = (n-1)*floor(n/3). - Laurance L. Y. Lau, Jun 22 2015

Mathematica

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 *)

Prog

(Small Basic)
b[2]=0
d[3]=2
d[4]=1
d[5]=1
For n=2 To 100
  If n+1 >=6 Then
    If Math.Remainder(n+1, 3)=0 Then
      d[n+1]=d[n-2]+4
    Else
      d[n+1]=d[n-2]+1
    EndIf
  EndIf
  b[n+1]=b[n]+d[n+1]
  TextWindow.Write(b[n]+", ")
EndFor
(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
(Magma) [(n-1)*Floor(n/3): n in [2..60]]; // Vincenzo Librandi, Jul 09 2015

Crossrefs

Cf. A074148 (Heart patterns), A229093 (Clubs patterns - fixed orientation), A229154 (Clubs Patterns - rotation allowed)

Sequence in context: A023725 A250051 A076079 * A196007 A134170 A276560

Adjacent sequences: A229590 A229591 A229592 * A229594 A229595 A229596

Keyword

nonn,easy

Author

Kival Ngaokrajang, Sep 26 2013

Extensions

G.f. adapted to the offset by Vincenzo Librandi, Oct 10 2013

Status

approved