

A229093


The clubs patterns appearing in n X n coins.


15



0, 0, 1, 2, 4, 6, 9, 12, 17, 22, 27, 34, 41, 48, 57, 66, 75, 86, 97, 108, 121, 134, 147, 162, 177, 192, 209, 226, 243, 262, 281, 300, 321, 342, 363, 386, 409, 432, 457, 482, 507, 534, 561, 588, 617, 646, 675, 706, 737, 768, 801, 834, 867, 902, 937, 972, 1009, 1046
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OFFSET

0,4


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. It is also A008810(n1), except for the third term. The inverse patterns (stars or voids between clubs) is A030511 (except the second term). See illustration in links.


LINKS



FORMULA

a(n) = ceiling((n1)^2/3), a(0) = 0, a(4) = 4.
G.f.: x^2*(x^72*x^6+x^5x^4+x^3x^21) / ((x1)^3*(x^2+x+1)).  Colin Barker, Oct 07 2013


MATHEMATICA

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 *)
LinearRecurrence[{2, 1, 1, 2, 1}, {0, 0, 1, 2, 4, 6, 9, 12, 17, 22}, 70] (* Harvey P. Dale, Feb 05 2020 *)


PROG

(PARI) Vec(x^2*(x^72*x^6+x^5x^4+x^3x^21)/((x1)^3*(x^2+x+1)) + O(x^100)) \\ Colin Barker, Oct 08 2013


CROSSREFS



KEYWORD

nonn,easy


AUTHOR



EXTENSIONS



STATUS

approved



