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%I #49 Jan 31 2022 19:40:55
%S 0,0,1,2,4,6,9,12,17,22,27,34,41,48,57,66,75,86,97,108,121,134,147,
%T 162,177,192,209,226,243,262,281,300,321,342,363,386,409,432,457,482,
%U 507,534,561,588,617,646,675,706,737,768,801,834,867,902,937,972,1009,1046
%N The clubs patterns appearing in n X n coins.
%C 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.
%C a(n) is the number of clubs patterns appearing in n X n coins. It is also A008810(n-1), except for the third term. The inverse patterns (stars or voids between clubs) is A030511 (except the second term). See illustration in links.
%H Vincenzo Librandi, <a href="/A229093/b229093.txt">Table of n, a(n) for n = 0..1000</a>
%H Kival Ngaokrajang, <a href="/A229093/a229093_1.pdf">Illustration for initial terms</a>
%H <a href="/index/Rec#order_05">Index entries for linear recurrences with constant coefficients</a>, signature (2,-1,1,-2,1).
%F a(n) = ceiling((n-1)^2/3), a(0) = 0, a(4) = 4.
%F G.f.: x^2*(x^7-2*x^6+x^5-x^4+x^3-x^2-1) / ((x-1)^3*(x^2+x+1)). - _Colin Barker_, Oct 07 2013
%t CoefficientList[Series[(x^7 - 2 x^6 + x^5 - x^4 + x^3 - x^2 - 1)/((x - 1)^3 (x^2 + x + 1)), {x, 0, 50}], x] (* _Vincenzo Librandi_, Oct 08 2013 *)
%t LinearRecurrence[{2,-1,1,-2,1},{0,0,1,2,4,6,9,12,17,22},70] (* _Harvey P. Dale_, Feb 05 2020 *)
%o (PARI) Vec(x^2*(x^7-2*x^6+x^5-x^4+x^3-x^2-1)/((x-1)^3*(x^2+x+1)) + O(x^100)) \\ _Colin Barker_, Oct 08 2013
%o (PARI) a(n) = ceil((n-1)^2/3) \\ _Charles R Greathouse IV_, Jan 06 2016
%Y Cf. A008810, A030511, A074148 (heart patterns), A227906, A229154.
%K nonn,easy
%O 0,4
%A _Kival Ngaokrajang_, Sep 13 2013
%E More terms from _Colin Barker_, Oct 08 2013
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