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A229154 The clubs patterns appearing in n X n coins, with rotation allowed. 11
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

From a japanese movie, 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 with rotation allowed. It is also A000212, except the fourth term. The inverse patterns (stars or voids between clubs) is A143978 (except 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.

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)). - Colin Barker, Oct 08 2013

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), A229093 (Clubs pattern fixed Orientation).

Sequence in context: A184430 A219657 A213707 * A174605 A108577 A272719

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 13 13:48 EDT 2020. Contains 335688 sequences. (Running on oeis4.)