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A231064 Coins left after packing X patterns into an n X n array of coins. 3
4, 4, 11, 15, 16, 24, 24, 31, 35, 41, 44, 49, 51, 55, 56, 64, 69, 71, 75, 76, 84, 89, 91, 95, 96, 104, 109, 111, 115, 116, 124, 129, 131, 135, 136, 144, 149, 151, 155, 156, 164, 169, 171, 175, 176, 184, 189, 191, 195, 196, 204, 209, 211, 215, 216, 224, 229, 231, 235, 236, 244, 249, 251 (list; graph; refs; listen; history; text; internal format)
OFFSET

2,1

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 total number of coins left (the coins out side X patterns) after packing X patterns into an n X n array of coins. The maximum number of X patterns that can be packed into an n X n array of coins is A231056 and voids left is A231065.

a(n) is also the total number of coins left after packing "+" patterns (8c5s1 type) into an n X n array of coins. See illustration in links.

LINKS

Table of n, a(n) for n=2..64.

Kival Ngaokrajang, Illustration of initial terms (U)

FORMULA

Empirical g.f.: x^2*(5*x^15 -5*x^14 -5*x^12 +5*x^11 -5*x^10 +5*x^9 +4*x^5 +x^4 +4*x^3 +7*x^2 +4) / ((x -1)^2*(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

  U = n*n - x[n]*5

  TextWindow.Write(U+", ")

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: A325859 A265206 A327684 * A302516 A254205 A048223

Adjacent sequences:  A231061 A231062 A231063 * A231065 A231066 A231067

KEYWORD

nonn

AUTHOR

Kival Ngaokrajang, Nov 03 2013

STATUS

approved

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Last modified May 29 20:42 EDT 2020. Contains 334710 sequences. (Running on oeis4.)