A229093 The clubs patterns appearing in n X n coins.

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

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(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.

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((n-1)^2/3), a(0) = 0, a(4) = 4.

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

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^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
(PARI) a(n) = ceil((n-1)^2/3) \\ Charles R Greathouse IV, Jan 06 2016

Crossrefs

Cf. A008810, A030511, A074148 (heart patterns), A227906, A229154.

Sequence in context: A338200 A194450 A080556 * A342371 A064985 A090631

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