A231065 -id:A231065 - cp's OEIS frontend

A231056 The maximum number of X patterns that can be packed into an n X n array of coins.

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

Offset: 2

Kival Ngaokrajang, Nov 03 2013

nonn

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.

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

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).

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

A231064 Coins left after packing X patterns into an n X n array of coins.

Original entry on oeis.org

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

Offset: 2

Kival Ngaokrajang, Nov 03 2013

nonn

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.

Kival Ngaokrajang, Illustration of initial terms (U)

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).

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

cp's OEIS Frontend

A231056 The maximum number of X patterns that can be packed into an n X n array of coins.

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