A229093 -id:A229093 - cp's OEIS frontend

A243608 Number T(n,k) of ways k L-tiles can be placed on an n X n square; triangle T(n,k), n>=0, 0<=k<=A229093(n), read by rows.

1, 1, 1, 1, 1, 4, 1, 1, 9, 20, 11, 1, 1, 16, 87, 196, 176, 46, 2, 1, 25, 244, 1195, 3145, 4431, 3161, 1007, 111, 2, 1, 36, 545, 4544, 22969, 73098, 147502, 185744, 140288, 59140, 12313, 1046, 26, 1, 49, 1056, 13215, 106819, 587149, 2251309, 6082000, 11562155

Offset: 0

Alois P. Heinz, Jun 07 2014

An L-tile is a 2 X 2 square with the upper right 1 X 1 subsquare removed and no rotations are allowed.

			T(3,1) = 4:
  ._____.   ._____.   ._____.   ._____.
  | |_|_|   |_|_|_|   |_| |_|   |_|_|_|
  |___|_|   | |_|_|   |_|___|   |_| |_|
  |_|_|_|   |___|_|   |_|_|_|   |_|___|
T(4,4) = 1:
  ._______.
  | |_| |_|
  |___|___|
  | |_| |_|
  |___|___|
T(5,6) = 2:
  ._________.   ._________.
  | |_|_| |_|   |_| |_| |_|
  |___| |___|   | |___|___|
  |_| |___|_|   |___|_| |_|
  | |___| |_|   | |_| |___|
  |___|_|___|   |___|___|_| .
Triangle T(n,k) begins:
  1;
  1;
  1,  1;
  1,  4,   1;
  1,  9,  20,   11,    1;
  1, 16,  87,  196,  176,   46,    2;
  1, 25, 244, 1195, 3145, 4431, 3161, 1007, 111, 2;

Alois P. Heinz, Rows n = 0..21, flattened

Columns k=0-6 give: A000012, A000290(n-1) for n>0, A243645, A243646, A243647, A243648, A243649.

Row sums give main diagonal of A226444 or A066864(n-1) for n>0.

b:= proc(n, l) option remember; local k;
      if n<2 then 1
    elif min(l[])>0 then b(n-1, map(h->h-1, l))
    else for k while l[k]>0 do od; expand(
         b(n, subsop(k=1, l))+ `if`(n>1 and k (p-> seq(coeff(p, x, i), i=0..degree(p)))(b(n, [0$n])):
seq(T(n), n=0..10);

b[n_, l_] := b[n, l] = Module[{k}, Which[n<2, 1, Min[l]>0, b[n-1, l-1], True, For[k = 1, l[[k]] > 0, k++]; Expand[b[n, ReplacePart[l, k -> 1]] + If[n>1 && k 2, k+1 -> 1}]], 0]]]];
T[n_] := Function[p, Table[Coefficient[p, x, i], {i, 0, Exponent[p, x]}]][ b[n, Table[0, {n}]]];
Table[T[n], {n, 0, 10}] // Flatten (* Jean-François Alcover, Apr 12 2017, translated from Maple *)

A229154 The clubs patterns appearing in n X n coins, with rotation allowed.

Original entry on oeis.org

1, 2, 5, 8, 12, 16, 21, 27, 33, 40, 48, 56, 65, 75, 85, 96, 108, 120, 133, 147, 161, 176, 192, 208, 225, 243, 261, 280, 300, 320, 341, 363, 385, 408, 432, 456, 481, 507, 533, 560, 588, 616, 645, 675, 705, 736, 768, 800, 833, 867, 901, 936, 972, 1008, 1045

Offset: 2

Kival Ngaokrajang, Sep 15 2013

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 clubs patterns appearing in n X n coins with rotation allowed. It is also A000212, except for the fourth term. The number of inverse patterns (stars or voids between clubs) is A143978 (except for the first term).

Vincenzo Librandi, Table of n, a(n) for n = 2..1000
Kival Ngaokrajang, Illustration of initial terms
Index entries for linear recurrences with constant coefficients, signature (2,-1,1,-2,1).

Cf. A000212, A143978, A074148 (Heart patterns), A227906, A229093 (Clubs pattern, fixed Orientation).

CoefficientList[Series[-(x^6 - 2 x^5 + x^4 - x^3 + 2 x^2 + 1)/((x - 1)^3 (x^2 + x + 1)), {x, 0, 60}], x] (* Vincenzo Librandi, Oct 08 2013 *)

Vec(-x^2*(x^6-2*x^5+x^4-x^3+2*x^2+1)/((x-1)^3*(x^2+x+1)) + O(x^100)) \\ Colin Barker, Oct 08 2013

a(n) = floor(n^2/3), a(3) = 2.
From Colin Barker, Oct 08 2013: (Start)
a(n) = n^2/3 + (2/9)*cos((2*Pi*n)/3) - 2/9.
G.f.: -x^2*(x^6-2*x^5+x^4-x^3+2*x^2+1) / ((x-1)^3*(x^2+x+1)). (End)

More terms from Colin Barker, Oct 08 2013

A229593 Number of boomerang patterns appearing in n X n coins, rotation not allowed.

Original entry on oeis.org

0, 2, 3, 4, 10, 12, 14, 24, 27, 30, 44, 48, 52, 70, 75, 80, 102, 108, 114, 140, 147, 154, 184, 192, 200, 234, 243, 252, 290, 300, 310, 352, 363, 374, 420, 432, 444, 494, 507, 520, 574, 588, 602, 660, 675, 690, 752, 768

Offset: 2

Kival Ngaokrajang, Sep 26 2013

The boomerang pattern is one of a total of 17 distinct patterns appearing in a 3 X 2 rectangular array of coins where each pattern consists of perimeter parts from each of 6 coins and forms a continuous area. See illustration of 6-curve patterns in links.
a(n) is the number of boomerang patterns appearing in an n X n array of coins with rotation not allowed. The number of inverse patterns is given in A229598.
It appears that a(n+1) is equivalent to n multiplied by the least possible number of addends in the partition in which the addends are multiplied together to produce the largest possible product for all n > 2. E.g., in the case of a(11), we look for partitions of 10, and for each partition we take the product of all its addends. The largest possible product formed is 3*3*2*2 = 3*3*4 = 36. The least possible number of addends here is 3, which we multiply by 10 to get 30. - Laurance L. Y. Lau, Jun 22 2015

Vincenzo Librandi, Table of n, a(n) for n = 2..1000
Kival Ngaokrajang, Illustration of initial terms
Kival Ngaokrajang, Illustration of 6-curve patterns
Index entries for linear recurrences with constant coefficients, signature (1,0,2,-2,0,-1,1).

Cf. A074148 (Heart patterns), A229093 (Clubs patterns - fixed orientation), A229154 (Clubs Patterns - rotation allowed)

[(n-1)*Floor(n/3): n in [2..60]]; // Vincenzo Librandi, Jul 09 2015

CoefficientList[Series[(2 x^4 + x^3 + x^2 + 2 x)/((1 - x^3)^2 (1 - x)), {x, 0, 80}], x] (* Vincenzo Librandi, Oct 10 2013 *)

a(n)=([0,1,0,0,0,0,0; 0,0,1,0,0,0,0; 0,0,0,1,0,0,0; 0,0,0,0,1,0,0; 0,0,0,0,0,1,0; 0,0,0,0,0,0,1; 1,-1,0,-2,2,0,1]^(n-2)*[0;2;3;4;10;12;14])[1,1] \\ Charles R Greathouse IV, Jun 16 2015

G.f.: (2*x^6 + x^5 + x^4 + 2*x^3)/((1-x^3)^2 * (1-x)). - Ralf Stephan, Oct 05 2013
3*a(n) = (1-n)^2 -2*A057078(n) +(-1)^n*A110665(n+1). - R. J. Mathar, Oct 09 2013
a(n) = (n-1)*floor(n/3). - Laurance L. Y. Lau, Jun 22 2015

G.f. adapted to the offset by Vincenzo Librandi, Oct 10 2013

A229598 Voids left when packing boomerangs into n X n coins.

Original entry on oeis.org

1, 0, 3, 8, 5, 12, 21, 16, 27, 40, 33, 48, 65, 56, 75, 96, 85, 108, 133, 120, 147, 176, 161, 192, 225, 208, 243, 280, 261, 300, 341, 320, 363, 408, 385, 432, 481, 456, 507, 560, 533, 588, 645, 616, 675, 736, 705, 768

Offset: 2

Kival Ngaokrajang, Sep 26 2013

nonn

The inverse patterns are voids or stars (consist of 4 perimeter parts of coins) appearing in n X n coins using the same rule as A229593.

			For n = 2, there is no boomerang can be packed into 2X2 coins, there is 1 void left, a(2) = 1.
For n = 3, there are 2 boomerangs can be packed into 3X3 coins with no void left, a(3) = 0.
...

Kival Ngaokrajang, Illustration of initial terms

Cf. A229093 (Illustration of inverse clubs patterns).

Empirical g.f.: -x^2*(x+1)*(x^4-2*x^3+5*x^2-2*x+1) / ((x-1)^3*(x^2+x+1)^2). - Colin Barker, Oct 06 2013

A230267 Coins left after packing 5 curves coins patterns into fountain of coins base n.

Original entry on oeis.org

1, 3, 2, 6, 7, 9, 12, 16, 17, 23, 26, 30, 35, 41, 44, 52, 57, 63, 70, 78, 83, 93, 100, 108, 117, 127, 134, 146, 155, 165, 176, 188, 197, 211, 222, 234, 247, 261, 272, 288, 301, 315, 330, 346, 359, 377, 392, 408, 425, 443

Offset: 1

Kival Ngaokrajang, Oct 15 2013

nonn

Refer to arrangement same as A005169: "A fountain is formed by starting with a row of coins, then stacking additional coins on top so that each new coin touches two in the previous row". The 5 curves coins patterns consist of a part of circumference and forms continuous area. There is total 13 distinct patterns. I would like to call "5C4S" type as it cover 4 coins and symmetry. When packing 5C4S into fountain of coins base n, the total number of 5C4S is A001399, the coins left is a(n) and void is A230276. See illustration in links.

Kival Ngaokrajang, Illustration of initial terms (U)

Cf. A008795 (3-curves coins patterns), A074148, A229093, A229154 (4-curves coins patterns), A001399 (5-curves coins patterns), A229593 (6-curves coins patterns).

G.f.: x*(x^3 - 2*x^2 + 2*x + 1)/((1-x)*(1-x^2)*(1-x^3)) (conjectured). - Ralf Stephan, Oct 17 2013

A230276 Voids left after packing 5-curves coins patterns into fountain of coins with base n.

Original entry on oeis.org

0, 1, 1, 6, 10, 16, 24, 34, 43, 57, 70, 85, 102, 121, 139, 162, 184, 208, 234, 262, 289, 321, 352, 385, 420, 457, 493, 534, 574, 616, 660, 706, 751, 801, 850, 901, 954, 1009, 1063, 1122, 1180, 1240, 1302, 1366, 1429

Offset: 1

Kival Ngaokrajang, Oct 15 2013

Refer to arrangement same as A005169: "A fountain is formed by starting with a row of coins, then stacking additional coins on top so that each new coin touches two in the previous row". The 5 curves coins patterns consist of a part of each coin circumference and forms a continuous area. There are total 13 distinct patterns. For selected pattern, I would like to call "5C4S" type as it cover 4 coins and symmetry. When packing 5C4S into fountain of coins base n, the total number of 5C4S is A001399, the coins left is A230267 and void left is a(n). See illustration in links.

Kival Ngaokrajang, Illustration of initial terms (V)
Index entries for linear recurrences with constant coefficients, signature (1,1,0,-1,-1,1).

Cf. A008795 (3-curves coins patterns), A074148, A229093, A229154 (4-curves coins patterns), A001399 (5-curves coins patterns), A229593 (6-curves coins patterns).

A099837 := proc(n)
    op(modp(n,3)+1,[2,-1,-1]) ;
end proc:
A230276 := proc(n)
    -A099837(n)/3 + (-48*n+31+18*n^2+9*(-1)^n)/24 ;
end proc:
seq(A230276(n),n=1..40) ; # R. J. Mathar, Feb 28 2018

LinearRecurrence[{1, 1, 0, -1, -1, 1}, {0, 1, 1, 6, 10, 16}, 45] (* Jean-François Alcover, May 05 2023 *)

G.f.: x^2*(x^4 + 3*x^3 + 4*x^2 + 1)/((1-x)*(1-x^2)*(1-x^3)). - Ralf Stephan, Oct 17 2013

a(n) = (9*(-1)^n+18*n^2-48*n)/24 - A099837(n)/3. - R. J. Mathar, Feb 28 2018

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

Kival Ngaokrajang, Oct 19 2013

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.

Kival Ngaokrajang, Illustration of initial terms (U)
Index entries for linear recurrences with constant coefficients, signature (0,2,0,-1).

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

With[{nn=60},Join[{2,4},Riffle[Range[4,nn,2],Range[9,2nn+1,4]]]] (* Harvey P. Dale, Feb 11 2015 *)

Vec(-x^2*(x^5-x^3-4*x-2)/((x-1)^2*(x+1)^2) + O(x^100)) \\ Colin Barker, Oct 30 2013

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)

A230370 Voids left after packing 3 curves coins patterns (3c3s type) into fountain of coins base n.

Original entry on oeis.org

0, 0, 3, 6, 13, 19, 39, 54, 66, 85, 100, 123, 141, 168, 189, 220, 244, 279, 306, 345, 375, 418, 451, 498, 534, 585, 624, 679, 721, 780, 825, 888, 936, 1003, 1054, 1125, 1179, 1254, 1311, 1390, 1450, 1533, 1596, 1683

Offset: 1

Kival Ngaokrajang, Oct 17 2013

nonn

Refer to arrangement same as A005169: "A fountain is formed by starting with a row of coins, then stacking additional coins on top so that each new coin touches two in the previous row". The 3 curves coins patterns consist of a part of each coin circumference and forms a continuous area. There are total 4 distinct patterns. For selected pattern, I would like to call "3c3s" type as it cover 3 coins and symmetry. When packing 3c3s into fountain of coins base n, the total number of 3c3s is A008805, the coins left is A008795 and voids left is a(n). See illustration in links.

Kival Ngaokrajang, Illustration of initial terms (V)

A001399, A230267, A230276 (5-curves coins patterns); A074148, A229093, A220154 (4-curves coins patterns); A008795 (3-curves coins patterns).

G.f.: x^3*(11*x^8 - 5*x^7 - 21*x^6 + 6*x^5 + 9*x^4 + x^2 + 3*x + 3)/((1-x)*(1-x^2)^2) (conjectured). Ralf Stephan, Oct 19 2013

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

Kival Ngaokrajang, Oct 23 2013

nonn

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.

Kival Ngaokrajang, Illustration of initial terms (T)

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

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

Kival Ngaokrajang, Oct 23 2013

nonn

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.

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 (6-curves).

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

cp's OEIS Frontend

A243608 Number T(n,k) of ways k L-tiles can be placed on an n X n square; triangle T(n,k), n>=0, 0<=k<=A229093(n), read by rows.

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A229154 The clubs patterns appearing in n X n coins, with rotation allowed.

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A229593 Number of boomerang patterns appearing in n X n coins, rotation not allowed.

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A229598 Voids left when packing boomerangs into n X n coins.

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A230267 Coins left after packing 5 curves coins patterns into fountain of coins base n.

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A230276 Voids left after packing 5-curves coins patterns into fountain of coins with base n.

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Maple

Mathematica

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

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Mathematica

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A230370 Voids left after packing 3 curves coins patterns (3c3s type) into fountain of coins base n.

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