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Search: a230267 -id:a230267
Displaying 1-9 of 9 results found.
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A230276
Voids left after packing 5-curves coins patterns into fountain of coins with base n.
+10
10
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
(list; graph; refs; listen; history; text; internal format)
OFFSET
1,4
COMMENTS
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.
LINKS
Table of n, a(n) for n=1..45.
Kival Ngaokrajang, Illustration of initial terms (V)
Index entries for linear recurrences with constant coefficients, signature (1,1,0,-1,-1,1).
FORMULA
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
MAPLE
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
MATHEMATICA
LinearRecurrence[{1, 1, 0, -1, -1, 1}, {0, 1, 1, 6, 10, 16}, 45] (* Jean-François Alcover, May 05 2023 *)
PROG
(Small Basic)
a[1]=0
d[2]=1
For n = 1 To 100
If n+1 >= 3 Then
If Math.Remainder(n+1, 3)=math.Remainder(n+1, 6) Then
d2=2
Else
If Math.Remainder(n+1, 3)+math.Remainder(n+1, 6)=5 then
d2=5
Else
d2=-1
EndIf
EndIf
d[n+1]=d[n]+d2
EndIf
a[n+1]=a[n]+d[n+1]
TextWindow.Write(a[n]+", ")
EndFor
CROSSREFS
Cf. A008795 (3-curves coins patterns), A074148, A229093, A229154 (4-curves coins patterns), A001399 (5-curves coins patterns), A229593 (6-curves coins patterns).
KEYWORD
nonn,easy
AUTHOR
Kival Ngaokrajang, Oct 15 2013
STATUS
approved
A227906
Coins left after packing heart patterns (fixed orientation) into n X n coins.
+10
9
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
(list; graph; refs; listen; history; text; internal format)
OFFSET
2,1
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
Table of n, a(n) for n=2..61.
Kival Ngaokrajang, Illustration of initial terms (U)
Index entries for linear recurrences with constant coefficients, signature (0,2,0,-1).
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)
MATHEMATICA
With[{nn=60}, Join[{2, 4}, Riffle[Range[4, nn, 2], Range[9, 2nn+1, 4]]]] (* Harvey P. Dale, Feb 11 2015 *)
PROG
(Small Basic)
For n = 2 To 100
If Math.Remainder(n, 2) = 0 then
a = n
Else
a = a + n
If n = 3 then
a = a - 1
endif
EndIf
TextWindow.Write(a+", ")
EndFor
(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
CROSSREFS
Cf. A008795, A230370 (3-curves), A074148, A229093, A229154 (4-curves), A001399, A230267, A230276 (5-curves), A229593, A228949, A229598 (6-curves).
KEYWORD
nonn,easy
AUTHOR
Kival Ngaokrajang, Oct 19 2013
STATUS
approved
A230370
Voids left after packing 3 curves coins patterns (3c3s type) into fountain of coins base n.
+10
8
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
(list; graph; refs; listen; history; text; internal format)
OFFSET
1,3
COMMENTS
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.
LINKS
Table of n, a(n) for n=1..44.
Kival Ngaokrajang, Illustration of initial terms (V)
FORMULA
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
PROG
(Small Basic)
a[1]=0
a[2]=0
d1[3]=3
For n=1 To 100
If n+2>=4 Then
If Math.Remainder(n+2, 2)=0 Then
d2= 2-(n+2)/2
Else
d2= (n+5)/2
EndIf
d1[n+2]=d1[n+1]+d2
EndIf
a[n+2]=a[n+1]+d1[n+2]
TextWindow.Write(a[n]+", ")
EndFor
CROSSREFS
A001399, A230267, A230276 (5-curves coins patterns); A074148, A229093, A220154 (4-curves coins patterns); A008795 (3-curves coins patterns).
KEYWORD
nonn
AUTHOR
Kival Ngaokrajang, Oct 17 2013
STATUS
approved
A230548
Twin hearts patterns packing into n X n coins.
+10
6
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
(list; graph; refs; listen; history; text; internal format)
OFFSET
2,3
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
Table of n, a(n) for n=2..52.
Kival Ngaokrajang, Illustration of initial terms (T)
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
PROG
(Small Basic)
col = 1
row = 0
For n = 2 To 100
add = 0
If Math.Remainder(n, 2) * Math.Remainder(n, 3) <> 0 Then
add = 1
EndIf
If n >= 4 And Math.Remainder(n, 2) = 0 Then
col = col + 1
EndIf
If n >= 3 And Math.Remainder(n, 3) = 0 Then
row = row + 1
EndIf
T = col * row + add
TextWindow.Write(T+", ")
EndFor
CROSSREFS
Cf. A008795, A230370 (3-curves); A074148, A227906, A229093, A229154 (4-curves); A001399, A230267, A230276 (5-curves); A229593, A228949, A229598, A002620 (6-curves).
KEYWORD
nonn
AUTHOR
Kival Ngaokrajang, Oct 23 2013
STATUS
approved
A230549
Coins left after packing twin hearts patterns into n X n coins.
+10
6
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
(list; graph; refs; listen; history; text; internal format)
OFFSET
2,1
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
Table of n, a(n) for n=2..49.
Kival Ngaokrajang, Illustration of initial terms (U)
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
PROG
(Small Basic)
col = 1
row = 0
For n = 2 To 100
add = 0
If Math.Remainder(n, 2) * Math.Remainder(n, 3) <> 0 Then
add = 1
EndIf
If n >= 4 And Math.Remainder(n, 2) = 0 Then
col = col + 1
EndIf
If n >= 3 And Math.Remainder(n, 3) = 0 Then
row = row + 1
EndIf
U = n * n - (col * row + add)*4
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 (6-curves).
KEYWORD
nonn
AUTHOR
Kival Ngaokrajang, Oct 23 2013
STATUS
approved
A230550
Voids left after packing twin hearts patterns into n X n coins.
+10
6
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
(list; graph; refs; listen; history; text; internal format)
OFFSET
2,2
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
Table of n, a(n) for n=2..47.
Kival Ngaokrajang, Illustration of initial terms (V)
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
PROG
(Small Basic)
col = 1
row = 0
For n = 2 To 100
add = 0
If Math.Remainder(n, 2) * Math.Remainder(n, 3) <> 0 Then
add = 1
EndIf
If n >= 4 And Math.Remainder(n, 2) = 0 Then
col = col + 1
EndIf
If n >= 3 And Math.Remainder(n, 3) = 0 Then
row = row + 1
EndIf
V = (n-1) * (n-1) - (col * row + add)*2
TextWindow.Write(V+", ")
EndFor
CROSSREFS
Cf. A008795, A230370 (3-curves); A074148, A227906, A229093, A229154 (4-curves); A001399, A230267, A230276 (5-curves); A229593, A228949, A229598, A002620 (6-curves).
KEYWORD
nonn
AUTHOR
Kival Ngaokrajang, Oct 23 2013
STATUS
approved
A231056
The maximum number of X patterns that can be packed into an n X n array of coins.
+10
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).
KEYWORD
nonn
AUTHOR
Kival Ngaokrajang, Nov 03 2013
STATUS
approved
A231064
Coins left after packing X patterns into an n X n array of coins.
+10
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).
KEYWORD
nonn
AUTHOR
Kival Ngaokrajang, Nov 03 2013
STATUS
approved
A231065
Voids left after packing X patterns into an of n X n array of coins.
+10
3
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
(list; graph; refs; listen; history; text; internal format)
OFFSET
2,3
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
Table of n, a(n) for n=2..63.
Kival Ngaokrajang, Illustration of initial terms (V)
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
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
V = (n-1)*(n-1) - x[n]*4
TextWindow.Write(V+", ")
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).
KEYWORD
nonn
AUTHOR
Kival Ngaokrajang, Nov 03 2013
STATUS
approved
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