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

2,2

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).

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)

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 *)

Prog

(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

Crossrefs

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

Sequence in context: A184430 A219657 A213707 * A362601 A174605 A108577

Adjacent sequences: A229151 A229152 A229153 * A229155 A229156 A229157

Keyword

nonn,easy

Author

Kival Ngaokrajang, Sep 15 2013

Extensions

More terms from Colin Barker, Oct 08 2013

Status

approved