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A231056 The maximum number of X patterns that can be packed into an n X n array of coins. 3
0, 1, 1, 2, 4, 5, 8, 10, 13, 16, 20, 24, 29, 34, 40, 45, 51, 58, 65, 73, 80, 88, 97, 106, 116, 125, 135, 146, 157, 169, 180, 192, 205, 218, 232, 245, 259, 274, 289, 305, 320, 336, 353, 370, 388, 405, 423, 442, 461, 481, 500, 520, 541, 562, 584, 605, 627, 650, 673, 697, 720, 744, 769, 794 (list; graph; refs; listen; history; text; internal format)
OFFSET

2,4

COMMENTS

The X pattern (8c5s2 type) is a pattern in which 8 curves cover 5 coins, and is one of a total of 13 such distinct patterns that appear in a tightly-packed 3 X 3 square array of coins of identical size; each of the 8 curves is a circular arc lying along the edge of one of the 5 coins, and the 8 curves are joined end-to-end to form a continuous area.

a(n) is the maximum number of X patterns that can be packed into an n X n array of coins. The total coins left after packing X patterns into an n X n array of coins is A231064 and voids left is A231065.

a(n) is also the maximum number of "+" patterns (8c5s1 type) that can be packed into an n X n array of coins. See illustration in links.

LINKS

Table of n, a(n) for n=2..65.

Kival Ngaokrajang, Illustration of initial terms (X and +)

FORMULA

Empirical g.f.: -x^3*(x^15 -2*x^14 +x^13 -x^12 +2*x^11 -2*x^10 +2*x^9 -x^8 +x^5 -x^4 +x^3 +x^2 -x +1) / ((x -1)^3*(x^4 +x^3 +x^2 +x +1)). - Colin Barker, Nov 27 2013

PROG

(Small Basic)

x[2] = 0

d1[3] = 1

For n = 2 To 100

  If Math.Remainder(n+2, 5) = 1 Then

     d2 = 0

  Else

    If Math.Remainder(n+2, 5) = 4 Then

     d2 = -1

    else

     d2 = 1

    EndIf

  EndIf

  d1[n+2] = d1[n+1] + d2

  x[n+1] = x[n] + d1[n+1]

  If n >= 13 And Math.Remainder(n, 5) = 3 Then

    x[n] = x[n] - 1

  EndIf

  If n=6 or n>=16 And Math.Remainder(n, 5)=1 Then

    x[n] = x[n] + 1

  EndIf

  TextWindow.Write(x[n]+", ")

EndFor

CROSSREFS

Cf. A008795, A230370 (3-curves); A074148, A227906, A229093, A229154 (4-curves); A001399, A230267, A230276 (5-curves); A229593, A228949, A229598, A002620, A230548, A230549, A230550 (6-curves).

Sequence in context: A227800 A144876 A239517 * A325363 A000549 A191985

Adjacent sequences:  A231053 A231054 A231055 * A231057 A231058 A231059

KEYWORD

nonn

AUTHOR

Kival Ngaokrajang, Nov 03 2013

STATUS

approved

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Last modified July 14 15:26 EDT 2020. Contains 335729 sequences. (Running on oeis4.)