

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

In the Japanese TV show "Tsuki no Koibito", the girl told her boyfriend that she saw a heart in 4 coins. Actually there are total 6 distinct patterns appearing in 2X2 coins which each pattern consist of a part of perimeter of each coin.
a(n) is the number of clubs patterns appearing in n X n coins. It is also A008810(n1), except the third term. The inverse patterns (stars or voids between clubs) is A030511 (except the second term). See illustration in links.


LINKS

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Kival Ngaokrajang, Illustration for initial terms
Index entries for linear recurrences with constant coefficients, signature (2,1,1,2,1).


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
(PARI) a(n) = ceil((n1)^2/3) \\ Charles R Greathouse IV, Jan 06 2016


CROSSREFS

Cf. A008810, A030511, A074148 (heart patterns).
Sequence in context: A080548 A194450 A080556 * A064985 A090631 A001365
Adjacent sequences: A229090 A229091 A229092 * A229094 A229095 A229096


KEYWORD

nonn,easy


AUTHOR

Kival Ngaokrajang, Sep 13 2013


EXTENSIONS

More terms from Colin Barker, Oct 08 2013


STATUS

approved



