A229154 The clubs patterns appearing in n X n coins, with rotation allowed.

1, 2, 5, 8, 12, 16, 21, 27, 33, 40, 48, 56, 65, 75, 85, 96, 108, 120, 133, 147, 161, 176, 192, 208, 225, 243, 261, 280, 300, 320, 341, 363, 385, 408, 432, 456, 481, 507, 533, 560, 588, 616, 645, 675, 705, 736, 768, 800, 833, 867, 901, 936, 972, 1008, 1045

Offset: 2

Author

Kival Ngaokrajang, Sep 15 2013

Keywords

• nonn
• easy

Comments

On the Japanese TV show "Tsuki no Koibito", a girl told her boyfriend that she saw a heart in 4 coins. Actually there are a total of 6 distinct patterns appearing in 2 X 2 coins in which each pattern consists of a part of the perimeter of each coin and forms a continuous area.

a(n) is the number of clubs patterns appearing in n X n coins with rotation allowed. It is also A000212, except for the fourth term. The number of inverse patterns (stars or voids between clubs) is A143978 (except for the first term).

Links

• Vincenzo Librandi, Table of n, a(n) for n = 2..1000
• Kival Ngaokrajang, Illustration of initial terms
• Index entries for linear recurrences with constant coefficients, signature (2,-1,1,-2,1).

Crossrefs

Cf. A000212, A143978, A074148 (Heart patterns), A227906, A229093 (Clubs pattern, fixed Orientation).

Programs

Mathematica

CoefficientList[Series[-(x^6 - 2 x^5 + x^4 - x^3 + 2 x^2 + 1)/((x - 1)^3 (x^2 + x + 1)), {x, 0, 60}], x] (* Vincenzo Librandi, Oct 08 2013 *)

PARI

Vec(-x^2*(x^6-2*x^5+x^4-x^3+2*x^2+1)/((x-1)^3*(x^2+x+1)) + O(x^100)) \\ Colin Barker, Oct 08 2013

Formula

a(n) = floor(n^2/3), a(3) = 2.

From Colin Barker, Oct 08 2013: (Start)

a(n) = n^2/3 + (2/9)*cos((2*Pi*n)/3) - 2/9.

G.f.: -x^2*(x^6-2*x^5+x^4-x^3+2*x^2+1) / ((x-1)^3*(x^2+x+1)). (End)

Extensions

More terms from Colin Barker, Oct 08 2013