login
The OEIS is supported by the many generous donors to the OEIS Foundation.

 

Logo
Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
A229593 Number of boomerang patterns appearing in n X n coins, rotation not allowed. 12
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 (list; graph; refs; listen; history; text; internal format)
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
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
KEYWORD
nonn,easy
AUTHOR
Kival Ngaokrajang, Sep 26 2013
EXTENSIONS
G.f. adapted to the offset by Vincenzo Librandi, Oct 10 2013
STATUS
approved

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recents
The OEIS Community | Maintained by The OEIS Foundation Inc.

License Agreements, Terms of Use, Privacy Policy. .

Last modified March 3 04:18 EST 2024. Contains 370499 sequences. (Running on oeis4.)