cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-7 of 7 results.

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

Original entry on oeis.org

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

Views

Author

Kival Ngaokrajang, Oct 19 2013

Keywords

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

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)

A230548 Twin hearts patterns packing into n X n coins.

Original entry on oeis.org

0, 1, 2, 3, 6, 7, 8, 12, 15, 16, 24, 25, 28, 35, 40, 41, 54, 55, 60, 70, 77, 78, 96, 97, 104, 117, 126, 127, 150, 151, 160, 176, 187, 188, 216, 217, 228, 247, 260, 261, 294, 295, 308, 330, 345, 346, 384, 385, 400, 425, 442
Offset: 2

Views

Author

Kival Ngaokrajang, Oct 23 2013

Keywords

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 twin hearts patterns (6c4a type: 6-curves cover 4 coins) packing into n X n coins with rotation not allowed. The total coins left after packing twin hearts patterns into n X n coins is A230549 and voids left is A230550. See illustration in links.

Links

Crossrefs

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

Formula

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

A230549 Coins left after packing twin hearts patterns into n X n coins.

Original entry on oeis.org

4, 5, 8, 13, 12, 21, 32, 33, 40, 57, 48, 69, 84, 85, 96, 125, 108, 141, 160, 161, 176, 217, 192, 237, 260, 261, 280, 333, 300, 357, 384, 385, 408, 473, 432, 501, 532, 533, 560, 637, 588, 669, 704, 705, 736, 825, 768, 861
Offset: 2

Views

Author

Kival Ngaokrajang, Oct 23 2013

Keywords

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 total coins 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 is A230548 and voids left is A230550. See illustration in links.

Links

Crossrefs

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

Formula

a(n) = n^2 - 4*A230548(n).
G.f.: x^2 * (-3*x^10 - 4*x^8 + 3*x^7 + 8*x^6 + 4*x^5 - x^4 + 4*x^3 + 4*x^2 + 5*x + 4)/(1+x^3)*(1-x^3)^2*(1-x^2). (conjectured). - Ralf Stephan, Oct 30 2013

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

Original entry on oeis.org

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

Views

Author

Kival Ngaokrajang, Oct 23 2013

Keywords

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

Crossrefs

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

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

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

Original entry on oeis.org

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

Views

Author

Kival Ngaokrajang, Nov 03 2013

Keywords

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

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).

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

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

Views

Author

Kival Ngaokrajang, Nov 03 2013

Keywords

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

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).

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

A231065 Voids left after packing X patterns into an of n X n array of coins.

Original entry on oeis.org

1, 0, 5, 8, 9, 16, 17, 24, 29, 36, 41, 48, 53, 60, 65, 76, 85, 92, 101, 108, 121, 132, 141, 152, 161, 176, 189, 200, 213, 224, 241, 256, 269, 284, 297, 316, 333, 348, 365, 380, 401, 420, 437, 456, 473, 496, 517, 536, 557, 576, 601, 624, 645, 668, 689, 716, 741, 764, 789, 812, 841, 868
Offset: 2

Views

Author

Kival Ngaokrajang, Nov 03 2013

Keywords

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 voids (spaces among coins) left 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 coins left is A231064.
a(n) is also the total number of voids left after packing "+" patterns (8c5s1 type) into an n X n array of coins. See illustration in links.

Links

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).

Formula

Empirical g.f.: x^2*(4*x^16 -8*x^15 +4*x^14 -4*x^13 +8*x^12 -8*x^11 +8*x^10 -4*x^9 +4*x^6 -5*x^5 +2*x^4 +2*x^3 -6*x^2 +2*x -1) / ((x -1)^3*(x^4 +x^3 +x^2 +x +1)). - Colin Barker, Nov 27 2013
Showing 1-7 of 7 results.

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