A227906 Coins left after packing heart patterns (fixed orientation) into n X n coins.

2, 4, 4, 9, 6, 13, 8, 17, 10, 21, 12, 25, 14, 29, 16, 33, 18, 37, 20, 41, 22, 45, 24, 49, 26, 53, 28, 57, 30, 61, 32, 65, 34, 69, 36, 73, 38, 77, 40, 81, 42, 85, 44, 89, 46, 93, 48, 97, 50, 101, 52, 105, 54, 109, 56, 113, 58, 117, 60, 121

Offset: 2

Author

Kival Ngaokrajang, Oct 19 2013

Keywords

• nonn
• easy

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 coins left after packing fixed orientation heart patterns (type 4c2s1: 4-curve cover 2 coins and symmetry) into n X n coins. The total number of hearts is A093005 and the number of voids left is A093353. See illustration in links.

Links

• Kival Ngaokrajang, Illustration of initial terms (U)
• Index entries for linear recurrences with constant coefficients, signature (0,2,0,-1).

Crossrefs

Cf. A008795, A230370 (3-curves), A074148, A229093, A229154 (4-curves), A001399, A230267, A230276 (5-curves), A229593, A228949, A229598 (6-curves).

Programs

Mathematica

With[{nn=60},Join[{2,4},Riffle[Range[4,nn,2],Range[9,2nn+1,4]]]] (* Harvey P. Dale, Feb 11 2015 *)

PARI

Vec(-x^2*(x^5-x^3-4*x-2)/((x-1)^2*(x+1)^2) + O(x^100)) \\ Colin Barker, Oct 30 2013

Formula

From Colin Barker, Oct 30 2013: (Start)

a(n) = (-1 + (-1)^n - (-3 + (-1)^n)*n)/2 for n>3.

a(n) = n for n>3 and even.

a(n) = 2*n-1 for n > 3 and odd.

a(n) = 2*a(n-2) - a(n-4) for n>7.

G.f.: -x^2*(x^5-x^3-4*x-2) / ((x-1)^2*(x+1)^2).(End)