A230550 Voids left after packing twin hearts patterns into n X n coins.

1, 2, 5, 10, 13, 22, 33, 40, 51, 68, 73, 94, 113, 126, 145, 174, 181, 214, 241, 260, 287, 328, 337, 382, 417, 442, 477, 530, 541, 598, 641, 672, 715, 780, 793, 862, 913, 950, 1001, 1078, 1093, 1174, 1233, 1276, 1335, 1424

Offset

2,2

Comments

Twin hearts (6c4a type) is one of total 17 distinct patterns appearing in 3X2 coins where each pattern consists of 6 perimeter parts from each coin and forms a continuous area.

a(n) is the number of total voids left after packing twin hearts patterns (6c4a type: 6-curves cover 4 coins) into n X n coins with rotation not allowed. The total twin hearts patterns packing into n X n coins is A230548 and coins left is A230549. See illustration in links.

Links

Table of n, a(n) for n=2..47.

Kival Ngaokrajang, Illustration of initial terms (V)

Formula

a(n) = (n-1)^2 - 2*A230548(n).

G.f.: x^2 * (-2*x^10 + x^9 + 2*x^8 + 8*x^7 + 11*x^6 + 8*x^5 + 6*x^4 + 7*x^3 + 4*x^2 + 2*x + 1)/((1+x^3)*(1-x^3)^2*(1-x^2)) (conjectured). - Ralf Stephan, Oct 30 2013

Prog

(Small Basic)
col = 1
row = 0
For n = 2 To 100
  add = 0
  If Math.Remainder(n, 2) * Math.Remainder(n, 3) <> 0 Then
    add = 1
  EndIf
  If n >= 4 And Math.Remainder(n, 2) = 0 Then
    col = col + 1
  EndIf
  If n >= 3 And Math.Remainder(n, 3) = 0 Then
    row = row + 1
  EndIf
  V = (n-1) * (n-1) - (col * row + add)*2
  TextWindow.Write(V+", ")
EndFor

Crossrefs

Cf. A008795, A230370 (3-curves); A074148, A227906, A229093, A229154 (4-curves); A001399, A230267, A230276 (5-curves); A229593, A228949, A229598, A002620 (6-curves).

Sequence in context: A190249 A188434 A135467 * A018571 A064233 A051952

Adjacent sequences: A230547 A230548 A230549 * A230551 A230552 A230553

Keyword

nonn

Author

Kival Ngaokrajang, Oct 23 2013

Status

approved