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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 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

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