A335613 -id:A335613 - cp's OEIS frontend

A340404 Number of sets in the geometry determined by the Hausdorff metric at each location between two sets defined by a complete bipartite graph K(5,n) (with n at least 4) missing three edges, where the removed edges are incident to different vertices in the 5-point set and none of the removed edges are incident to the same vertex in the other set.

66914, 2633546, 91075250, 2972550458, 94442228354, 2962494091946, 92361814692050, 2871102810997658, 89122619177691554, 2764578611249652746, 85728604753979332850, 2657986810333194788858, 82403592385902351106754, 2554601385194824465949546

Offset: 4

Rachel Wofford, Jan 06 2021

Start with a complete bipartite graph K(5,n) with vertex sets A and B where |A| = 5 and |B| is at least 4. We can arrange the points in sets A and B such that h(A,B) = d(a,b) for all a in A and b in B, where h is the Hausdorff metric. The pair [A,B] is a configuration. Then a set C is between A and B at location s if h(A,C) = h(C,B) = h(A,B) and h(A,C) = s. Call a pair ab, where a is in A and b is in B an edge. This sequence provides the number of sets between sets A' and B' at location s in a new configuration [A',B'] obtained from [A,B] by removing three edges, where the removed edges are incident to different points in A and none of the removed edges are incident to the same point in B. So this sequence tells the number of sets at each location on the line segment between A' and B'.
Number of {0,1} 5 X n matrices with three fixed zero entries none of which are in the same row or column with no zero rows or columns.
Take a complete bipartite graph K(5,n) (with n at least 4) having parts A and B where |A| = 5. This sequence gives the number of edge covers of the graph obtained from this K(5,n) graph after removing three edges, where the removed edges are incident to different vertices in A and none of the removed edges are incident to the same vertex in B.

Steven Schlicker, Roman Vasquez, and Rachel Wofford, Integer Sequences from Configurations in the Hausdorff Metric Geometry via Edge Covers of Bipartite Graphs, J. Int. Seq. (2023) Vol. 26, Art. 23.6.6.
Index entries for linear recurrences with constant coefficients, signature (57,-1002,6562,-15381,9765).

Sequences of segments from removing edges from bipartite graphs A335608-A335613, A337416-A337418, A340173-A340175, A340199-A340201, A340897-A340899, A342580, A342796, A342850, A340403-A340405, A340433-A340438, A341551-A341553, A342327-A342328, A343372-A343374, A343800. Polygonal chain sequences A152927, A152928, A152929, A152930, A152931, A152932, A152933, A152934, A152939. Number of {0,1} n X n matrices with no zero rows or columns A048291.

a(n) = 3375*31^(n-3) - 2891*15^(n-3) + 846*7^(n-3) - 10*3^(n-1) + 2.
From Alejandro J. Becerra Jr., Feb 13 2021: (Start)
G.f.: 2*x^4*(6064065*x^4 - 9507684*x^3 + 4005478*x^2 - 590276*x + 33457)/((1 - x)*(1 - 3*x)*(1 - 7*x)*(1 - 15*x)*(1 - 31*x)).
a(n) = 57*a(n-1) - 1002*a(n-2) + 6562*a(n-3) - 15381*a(n-4) + 9765*a(n-5). (End)

A340434 Number of sets in the geometry determined by the Hausdorff metric at each location between two sets defined by a complete bipartite graph K(5,n) (with n at least 4) missing three edges, where all three removed edges are incident to different vertices in the 5-point set but all three removed edges are incident to the same vertex in the other set.

Original entry on oeis.org

57152, 2248976, 77784248, 2538972344, 80670722672, 2530567728416, 78896594655848, 2452545943242824, 76130313033068192, 2361561349263377456, 73231232199903981848, 2270510693649412710104, 70390969213953161845712, 2182197113248136520812096, 67649266538598993456642248

Offset: 4

Rachel Wofford, Jan 07 2021

Start with a complete bipartite graph K(5,n) with vertex sets A and B where |A| = 5 and |B| is at least 4. We can arrange the points in sets A and B such that h(A,B) = d(a,b) for all a in A and b in B, where h is the Hausdorff metric. The pair [A,B] is a configuration. Then a set C is between A and B at location s if h(A,C) = h(C,B) = h(A,B) and h(A,C) = s. Call a pair ab, where a is in A and b is in B an edge. This sequence provides the number of sets between sets A' and B' at location s in a new configuration [A',B'] obtained from [A,B] by removing three edges, where all three removed edges are incident to different points in A but all three removed edges are incident to the same point in B. So this sequence tells the number of sets at each location on the line segment between A' and B'.
Number of {0,1} 5 X n matrices (with n at least 4) with three fixed zero entries all of which are in the same column with no zero rows or columns.
Take a complete bipartite graph K(5,n) (with n at least 4) having parts A and B where |A| = 5. This sequence gives the number of edge covers of the graph obtained from this K(5,n) graph after removing three edges, where the removed edges are incident to different vertices in A and none of the removed edges are incident to the same vertex in B.

Steven Schlicker, Roman Vasquez, and Rachel Wofford, Integer Sequences from Configurations in the Hausdorff Metric Geometry via Edge Covers of Bipartite Graphs, J. Int. Seq. (2023) Vol. 26, Art. 23.6.6.

Sequences of segments from removing edges from bipartite graphs A335608-A335613, A337416-A337418, A340173-A340175, A340199-A340201, A340897-A340899, A342580, A342796, A342850, A340403-A340405, A340433-A340438, A341551-A341553, A342327-A342328, A343372-A343374, A343800. Polygonal chain sequences A152927, A152928, A152929, A152930, A152931, A152932, A152933, A152934, A152939. Number of {0,1} n X n matrices with no zero rows or columns A048291.

a(n) = 3*31^(n-1) - 11*15^(n-1) + 15*7^(n-1) - 3^(n+1) + 2.

A340435 Number of sets in the geometry determined by the Hausdorff metric at each location between two sets defined by a complete bipartite graph K(6,n) (with n at least 4) missing three edges, where all three removed edges are incident to different vertices in the 6-point set but all three removed edges are incident to the same vertex in the other set.

Original entry on oeis.org

1014458, 85045184, 6126191066, 411615281120, 26756164505978, 1711634190402944, 108645469309633946, 6869952591587660960, 433592445465504586298, 27340695032306205303104, 1723219625722022031240026, 108586272500880714880450400, 6841661762270647247773447418

Offset: 4

Rachel Wofford, Jan 07 2021

Start with a complete bipartite graph K(6,n) with vertex sets A and B where |A| = 6 and |B| is at least 4. We can arrange the points in sets A and B such that h(A,B) = d(a,b) for all a in A and b in B, where h is the Hausdorff metric. The pair [A,B] is a configuration. Then a set C is between A and B at location s if h(A,C) = h(C,B) = h(A,B) and h(A,C) = s. Call a pair ab, where a is in A and b is in B an edge. This sequence provides the number of sets between sets A' and B' at location s in a new configuration [A',B'] obtained from [A,B] by removing three edges, where all three removed edges are incident to different points in A but all three removed edges are incident to the same point in B. So this sequence tells the number of sets at each location on the line segment between A' and B'.
Number of {0,1} 6 X n matrices (with n at least 4) with three fixed zero entries all of which are in the same column with no zero rows or columns.
Take a complete bipartite graph K(6,n) (with n at least 4) having parts A and B where |A| = 6. This sequence gives the number of edge covers of the graph obtained from this K(6,n) graph after removing three edges, where the removed edges are incident to different vertices in A and none of the removed edges are incident to the same vertex in B.

Steven Schlicker, Roman Vasquez, and Rachel Wofford, Integer Sequences from Configurations in the Hausdorff Metric Geometry via Edge Covers of Bipartite Graphs, J. Int. Seq. (2023) Vol. 26, Art. 23.6.6.

Sequences of segments from removing edges from bipartite graphs A335608-A335613, A337416-A337418, A340173-A340175, A340199-A340201, A340897-A340899, A342580, A342796, A342850, A340403-A340405, A340433-A340438, A341551-A341553, A342327-A342328, A343372-A343374, A343800. Polygonal chain sequences A152927, A152928, A152929, A152930, A152931, A152932, A152933, A152934, A152939. Number of {0,1} n X n matrices with no zero rows or columns A048291.

a(n) = 7*63^(n-1) - 30*31^(n-1) + 51*15^(n-1) - 43*7^(n-1) + 6*3^(n) - 3.

A340436 Number of sets in the geometry determined by the Hausdorff metric at each location between two sets defined by a complete bipartite graph K(3,n) (with n at least 4) missing three edges, where exactly two of the removed edges are incident to the same vertex in the 3-point set but none of the removed edges are incident to the same vertex in the other set.

Original entry on oeis.org

151, 1207, 8911, 63775, 450631, 3167047, 22207231, 155564335, 1089291511, 7626064087, 53385519151, 373707845695, 2615982554791, 18311960788327, 128183974232671, 897288565771855, 6281022198832471, 43967162107115767, 307770154895675791

Offset: 4

Rachel Wofford, Apr 02 2021

Start with a complete bipartite graph K(3,n) with vertex sets A and B where |A| = 3 and |B| is at least 4. We can arrange the points in sets A and B such that h(A,B) = d(a,b) for all a in A and b in B, where h is the Hausdorff metric. The pair [A,B] is a configuration. Then a set C is between A and B at location s if h(A,C) = h(C,B) = h(A,B) and h(A,C) = s. Call a pair ab, where a is in A and b is in B an edge. This sequence provides the number of sets between sets A' and B' at location s in a new configuration [A',B'] obtained from [A,B] by removing three edges, where exactly two of the removed edges are incident to the same point in A but none of the removed edges are incident to the same point in B. So this sequence tells the number of sets at each location on the line segment between A' and B'.
Number of {0,1} 3 X n matrices (with n at least 4) with three fixed zero entries where exactly two zero entries occur in one row and no column has more than one zero entry, with no zero rows or columns.
Take a complete bipartite graph K(3,n) (with n at least 4) having parts A and B where |A| = 3. This sequence gives the number of edge covers of the graph obtained from this K(3,n) graph after removing three edges, where exactly two of the removed edges are incident to the same vertex in A but none of the removed edges are incident to the same vertex in B.

Steven Schlicker, Roman Vasquez, and Rachel Wofford, Integer Sequences from Configurations in the Hausdorff Metric Geometry via Edge Covers of Bipartite Graphs, J. Int. Seq. (2023) Vol. 26, Art. 23.6.6.
Index entries for linear recurrences with constant coefficients, signature (11,-31,21).

Sequences of segments from removing edges from bipartite graphs A335608-A335613, A337416-A337418, A340173-A340175, A340199-A340201, A340897-A340899, A342580, A342796, A342850, A340403-A340405, A340433-A340438, A341551-A341553, A342327-A342328, A343372-A343374, A343800. Polygonal chain sequences A152927, A152928, A152929, A152930, A152931, A152932, A152933, A152934, A152939. Number of {0,1} n X n matrices with no zero rows or columns A048291.

a(n) = 27*7^(n-3) - 13*3^(n-3) + 1.

G.f.: x^4*(151 - 454*x + 315*x^2)/(1 - 11*x + 31*x^2 - 21*x^3). - Stefano Spezia, Apr 10 2021

A340437 Number of sets in the geometry determined by the Hausdorff metric at each location between two sets defined by a complete bipartite graph K(4,n) (with n at least 4) missing three edges, where exactly two of the removed edges are incident to the same vertex in the 4-point set but none of the removed edges are incident to the same vertex in the other set.

Original entry on oeis.org

3451, 64705, 1068475, 16734721, 256041211, 3875962945, 58387463995, 877549918081, 13175419936891, 197716513589185, 2966344254717115, 44499339824298241, 667519329831422971, 10012994576066466625, 150196351045142283835, 2252955292518089539201, 33794399575693844931451

Offset: 4

Rachel Wofford, Apr 02 2021

Start with a complete bipartite graph K(4,n) with vertex sets A and B where |A| = 4 and |B| is at least 4. We can arrange the points in sets A and B such that h(A,B) = d(a,b) for all a in A and b in B, where h is the Hausdorff metric. The pair [A,B] is a configuration. Then a set C is between A and B at location s if h(A,C) = h(C,B) = h(A,B) and h(A,C) = s. Call a pair ab, where a is in A and b is in B an edge. This sequence provides the number of sets between sets A' and B' at location s in a new configuration [A',B'] obtained from [A,B] by removing three edges, where exactly two of the removed edges are incident to the same point in A but none of the removed edges are incident to the same point in B. So this sequence tells the number of sets at each location on the line segment between A' and B'.
Number of {0,1} 4 X n matrices (with n at least 4) with three fixed zero entries where exactly two zero entries occur in one row and no column has more than one zero entry, with no zero rows or columns.
Take a complete bipartite graph K(4,n) (with n at least 4) having parts A and B where |A| = 4. This sequence gives the number of edge covers of the graph obtained from this K(4,n) graph after removing three edges, where exactly two of the removed edges are incident to the same vertex in A but none of the removed edges are incident to the same vertex in B.

Steven Schlicker, Roman Vasquez, and Rachel Wofford, Integer Sequences from Configurations in the Hausdorff Metric Geometry via Edge Covers of Bipartite Graphs, J. Int. Seq. (2023) Vol. 26, Art. 23.6.6.
Index entries for linear recurrences with constant coefficients, signature (26,-196,486,-315).

Sequences of segments from removing edges from bipartite graphs A335608-A335613, A337416-A337418, A340173-A340175, A340199-A340201, A340897-A340899, A342580, A342796, A342850, A340403-A340405, A340433-A340438, A341551-A341553, A342327-A342328, A343372-A343374, A343800. Polygonal chain sequences A152927, A152928, A152929, A152930, A152931, A152932, A152933, A152934, A152939. Number of {0,1} n X n matrices with no zero rows or columns A048291.

LinearRecurrence[{26,-196,486,-315},{3451,64705,1068475,16734721},20] (* Harvey P. Dale, Apr 03 2022 *)

a(n) = 343*15^(n-3) - 264*7^(n-3) + 52*3^(n-3) - 2.

G.f.: x^4*(3451 - 25021*x + 62541*x^2 - 40635*x^3)/(1 - 26*x + 196*x^2 - 486*x^3 + 315*x^4). - Stefano Spezia, Apr 10 2021

A340898 Number of sets in the geometry determined by the Hausdorff metric at each location between two sets defined by a complete bipartite graph K(3,n) (with n at least 4) missing three edges, where all three removed edges are incident to the same vertex in the 3-point set.

Original entry on oeis.org

104, 1064, 8480, 62480, 446744, 3155384, 22172240, 155459360, 1088976584, 7625119304, 53382684800, 373699342640, 2615957045624, 18311884260824, 128183744650160, 897287877024320, 6281020132589864, 43967155908387944, 307770136299492320

Offset: 4

Roman I. Vasquez, Jan 25 2021

Start with a complete bipartite graph K(3,n) with vertex sets A and B where |A| = 3 and |B| is at least 4. We can arrange the points in sets A and B such that h(A,B) = d(a,b) for all a in A and b in B, where h is the Hausdorff metric. The pair [A,B] is a configuration. Then a set C is between A and B at location s if h(A,C) = h(C,B) = h(A,B) and h(A,C) = s. Call a pair ab, where a is in A and b is in B an edge. This sequence provides the number of sets between sets A' and B' at location s in a new configuration [A',B'] obtained from [A,B] by removing three edges, where all three removed edges are incident to the same point in A. So this sequence gives the number of sets at each location on the line segment between A' and B'.
Number of {0,1} 3 X n matrices (with n at least 4) with three fixed zero entries all in the same row and no zero rows or columns.
Take a complete bipartite graph K(3,n) (with n at least 4) having parts A and B where |A| = 3. This sequence gives the number of edge covers of the graph obtained from this K(3,n) graph after removing three edges, where all three removed edges are incident same vertex in A.

Steven Schlicker, Roman Vasquez, and Rachel Wofford, Integer Sequences from Configurations in the Hausdorff Metric Geometry via Edge Covers of Bipartite Graphs, J. Int. Seq. (2023) Vol. 26, Art. 23.6.6.
Index entries for linear recurrences with constant coefficients, signature (11,-31,21).

Sequences of segments from removing edges from bipartite graphs A335608-A335613, A337416-A337418, A340173-A340175, A340199-A340201, A340897-A340899, A342580, A342796, A342850, A340403-A340405, A340433-A340438, A341551-A341553, A342327-A342328, A343372-A343374, A343800. Polygonal chain sequences A152927, A152928, A152929, A152930, A152931, A152932, A152933, A152934, A152939. Number of {0,1} n X n matrices with no zero rows or columns A048291.

a(n) = 27*7^(n-3) - 29*3^(n-3) + 2.
From Stefano Spezia, Jan 26 2021: (Start)
G.f.: 8*x^4*(13 - 10*x)/(1 - 11*x + 31*x^2 - 21*x^3).
a(n) = 11*a(n-1) - 31*a(n-2) + 21*a(n-3) for n > 6. (End)

A341552 Number of sets in the geometry determined by the Hausdorff metric at each location between two sets defined by a complete bipartite graph K(3,n) (with n at least 4) missing three edges, where all three removed edges are incident to different vertices in the 3 point set but exactly two removed edges are incident to the same vertex in the other set.

Original entry on oeis.org

129, 975, 7041, 49935, 351489, 2466255, 17281281, 121021455, 847307649, 5931625935, 41522798721, 290663842575, 2034659652609, 14242655832015, 99698705615361, 697891283681295, 4885240018890369, 34196683231596495, 239376791919267201, 1675637571329145615

Offset: 4

Steven Schlicker, Feb 14 2021

Start with a complete bipartite graph K(3,n) with vertex sets A and B where |A| = 3 and |B| is at least 4. We can arrange the points in sets A and B such that h(A,B) = d(a,b) for all a in A and b in B, where h is the Hausdorff metric. The pair [A,B] is a configuration. Then a set C is between A and B at location s if h(A,C) = h(C,B) = h(A,B) and h(A,C) = s. Call a pair ab, where a is in A and b is in B an edge. This sequence provides the number of sets between sets A' and B' at location s in a new configuration [A',B'] obtained from [A,B] by removing three edges, where all three removed edges are incident to different points in A but exactly two removed edges are incident to the same point in B. So this sequence tells the number of sets at each location on the line segment between A' and B'.
Number of {0,1} 3 X n matrices (with n at least 4) with three fixed zero entries where exactly two zero entries occur in one column and no row has more than one zero entry, with no zero rows or columns.
Take a complete bipartite graph K(3,n) (with n at least 4) having parts A and B where |A| = 3. This sequence gives the number of edge covers of the graph obtained from this K(3,n) graph after removing three edges, where all three removed edges are incident to different vertices in A but exactly two removed edges are incident to the same vertex in B.

Steven Schlicker, Roman Vasquez, and Rachel Wofford, Integer Sequences from Configurations in the Hausdorff Metric Geometry via Edge Covers of Bipartite Graphs, J. Int. Seq. (2023) Vol. 26, Art. 23.6.6.
Index entries for linear recurrences with constant coefficients, signature (10,-21).

Sequences of segments from removing edges from bipartite graphs A335608-A335613, A337416-A337418, A340173-A340175, A340199-A340201, A340897-A340899, A342580, A342796, A342850, A340403-A340405, A340433-A340438, A341551-A341553, A342327-A342328, A343372-A343374, A343800. Polygonal chain sequences A152927, A152928, A152929, A152930, A152931, A152932, A152933, A152934, A152939. Number of {0,1} n X n matrices with no zero rows or columns A048291.

a(n) = 3*7^(n-2)-2*3^(n-2).

G.f.: 3*x^4*(43 - 105*x)/(1 - 10*x + 21*x^2). - Stefano Spezia, Feb 14 2021

A343373 Number of sets in the geometry determined by the Hausdorff metric at each location between two sets defined by a complete bipartite graph K(4,n) (with n at least 4) missing three edges, where exactly two removed edges are incident to the same vertex in the 4-point set and exactly two removed edges are incident to the same vertex in the other set.

Original entry on oeis.org

3112, 58984, 978064, 15345952, 234980152, 3558436504, 53613281824, 805858151632, 12099490097992, 181573692295624, 2724174818398384, 40866608458275712, 613027030583891032, 9195600786027620344

Offset: 4

Steven Schlicker, Apr 12 2021

Start with a complete bipartite graph K(4,n) with vertex sets A and B where |A| = 4 and |B| is at least 4. We can arrange the points in sets A and B such that h(A,B) = d(a,b) for all a in A and b in B, where h is the Hausdorff metric. The pair [A,B] is a configuration. Then a set C is between A and B at location s if h(A,C) = h(C,B) = h(A,B) and h(A,C) = s. Call a pair ab, where a is in A and b is in B an edge. This sequence provides the number of sets between sets A' and B' at location s in a new configuration [A',B'] obtained from [A,B] by removing three edges, where exactly two removed edges are incident to the same point in A and exactly two removed edges are incident to the same point in B. So this sequence tells the number of sets at each location on the line segment between A' and B'.
Number of {0,1} 4 X n matrices (with n at least 4) with three fixed zero entries where exactly two zero entries occur in one row and exactly two zero entries occur in one column, with no zero rows or columns.
Take a complete bipartite graph K(4,n) (with n at least 4) having parts A and B where |A| = 4. This sequence gives the number of edge covers of the graph obtained from this K(4,n) graph after removing three edges, where exactly two removed edges are incident to the same vertex in A and exactly two removed edges are incident to the same vertex in B.

Steven Schlicker, Roman Vasquez, and Rachel Wofford, Integer Sequences from Configurations in the Hausdorff Metric Geometry via Edge Covers of Bipartite Graphs, J. Int. Seq. (2023) Vol. 26, Art. 23.6.6.
Index entries for linear recurrences with constant coefficients, signature (26,-196,486,-315).

Sequences of segments from removing edges from bipartite graphs A335608-A335613, A337416-A337418, A340173-A340175, A340199-A340201, A340897-A340899, A342580, A342796, A342850, A340403-A340405, A340433-A340438, A341551-A341553, A342327-A342328, A343372-A343374, A343800. Polygonal chain sequences A152927, A152928, A152929, A152930, A152931, A152932, A152933, A152934, A152939. Number of {0,1} n X n matrices with no zero rows or columns A048291.

a(n) = 21*15^(n-2) - 36*7^(n-2) + 17*3^(n-2) - 2.

G.f.: 8*x^4*(389 - 2741*x + 6804*x^2 - 4410*x^3)/(1 - 26*x + 196*x^2 - 486*x^3 + 315*x^4). - Stefano Spezia, Apr 13 2021

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A340898 Number of sets in the geometry determined by the Hausdorff metric at each location between two sets defined by a complete bipartite graph K(3,n) (with n at least 4) missing three edges, where all three removed edges are incident to the same vertex in the 3-point set.

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