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

Author

Kival Ngaokrajang, Sep 26 2013

Keywords

• nonn
• easy

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).

Crossrefs

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

Programs

Magma

[(n-1)*Floor(n/3): n in [2..60]]; // Vincenzo Librandi, Jul 09 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 *)

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

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

Extensions

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