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A229154 The clubs patterns appearing in n X n coins, with rotation allowed. 12
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 (list; graph; refs; listen; history; text; internal format)
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
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
KEYWORD
nonn,easy
AUTHOR
Kival Ngaokrajang, Sep 15 2013
EXTENSIONS
More terms from Colin Barker, Oct 08 2013
STATUS
approved

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Last modified July 14 05:06 EDT 2024. Contains 374291 sequences. (Running on oeis4.)