A340405 -id:A340405 - cp's OEIS frontend

A340897 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 3) missing two edges, where the two removed edges are not incident to the same vertex in the 6-point set and are also not incident to the same vertex in the other set.

21043, 2338345, 190200379, 13516195777, 902364046723, 58476376861465, 3735244109884939, 236920394417284657, 14975763121178295763, 945018874264393643785, 59584148902740043271899, 3755288737092394598648737, 236629307506201555636890403

Offset: 3

Roman I. Vasquez, Jan 25 2021

Start with a complete bipartite graph K(6,n) with vertex sets A and B where |A| = 6 and |B| is at least 3. 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 two edges, where the two removed edges are not incident to the same point in A and are also not incident to the same point in B. So this sequence gives 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 3) with two fixed zero entries not in the same row or column and no zero rows or columns.
Take a complete bipartite graph K(6,n) (with n at least 3) 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 two edges, where the two removed edges are not incident to the same vertex in A and are also not 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) = 961*63^(n-2) - 1830*31^(n-2) + 1359*15^(n-2) - 484*7^(n-2) + 79*3^(n-2) - 4.

A341551 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 exactly two of the removed edges are incident to the same vertex in the 6-point set but none of the removed edges are incident to the same vertex in the other set.

Original entry on oeis.org

996787, 87880249, 6458329435, 437811072433, 28577902283587, 1831839463314409, 116388761878654315, 7363089071153371873, 464825043098493809107, 29313469954934882953369, 1847663299656911486659195, 116431149842916469716759313, 7336041758469840870854326627

Offset: 4

Steven Schlicker, Feb 14 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 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} 6 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(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 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 (120,-4593,69688,-428787,978768,-615195).

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)= 29791*63^(n-3) - 34890*31^(n-3) + 14673*15^(n-3) - 2740*7^(n-3) + 211*3^(n-3) - 4.

A341553 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 all three removed edges are incident to different vertices in the 4-point set but exactly two removed edges are incident to the same vertex in the other set.

Original entry on oeis.org

3451, 61567, 996787, 15478951, 235916971, 3565011727, 53659360867, 806180862391, 12101749545691, 181589509846687, 2724285545507347, 40867383560793031, 613032456339776011, 9195638766433606447, 137935644948388268227, 2069042118396589446871

Offset: 4

Steven Schlicker, Mar 08 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 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} 4 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(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 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 (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.

Array[21*15^(# - 2) - 4*7^(# - 1) + 11*3^(# - 2) - 1 &, 16, 4] (* Michael De Vlieger, Mar 19 2021 *)

a(n) = 21*15^(n-2) - 4*7^(n-1) + 11*3^(n-2) - 1.

G.f.: x^4*(3451 - 28159*x + 72441*x^2 - 47565*x^3)/(1 - 26*x + 196*x^2 - 486 *x^3 + 315*x^4). - Stefano Spezia, Mar 08 2021

A342327 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 exactly two removed edges are incident to the same vertex in the other set.

Original entry on oeis.org

64705, 2542687, 87880249, 2867519047, 91094247025, 2857310964847, 89080092692329, 2769052985833687, 85954322576134945, 2666290098653287807, 82680590830861862809, 2563482326383161959527, 79473712585542654112465, 2463771499324688282695567

Offset: 4

Steven Schlicker, Mar 08 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 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} 5 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(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 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 (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.

Array[105*31^(# - 2) - 185*15^(# - 2) + 116*7^(# - 2) - 29*3^(# - 2) + 2 &, 14, 4] (* Michael De Vlieger, Mar 19 2021 *)

a(n) = 105*31^(n-2) - 185*15^(n-2) + 116*7^(n-2) - 29*3^(n-2) + 2.

A342328 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 exactly two removed edges are incident to the same vertex in the other set.

Original entry on oeis.org

1068475, 89633839, 6458329435, 433976684431, 28211055010555, 1804746233554159, 114556965257054875, 7243790885015626831, 457188176014823960635, 28828588756092946562479, 1816999192589895468925915, 114495695622871975031439631

Offset: 4

Steven Schlicker, Mar 08 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 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} 6 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(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 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 (120,-4593,69688,-428787,978768,-615195).

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.

Array[465*63^(# - 2) - 982*31^(# - 2) + 807*15^(# - 2) - 316*7^(# - 2) + 56*3^(# - 2) - 3 &, 12, 4] (* Michael De Vlieger, Mar 19 2021 *)

a(n) = 465*63^(n-2) - 982*31^(n-2) + 807*15^(n-2) - 316*7^(n-2) + 56*3^(n-2) - 3.

A342580 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 the same vertex in the 5-point set.

Original entry on oeis.org

43664, 2248976, 85045184, 2880236192, 93044373104, 2941433979056, 92045266123424, 2866350051682112, 89051296064477264, 2763508542463136336, 85712552167491668864, 2657746010652834993632, 82399980314514994098224, 2554547203590738451564016

Offset: 4

Roman I. Vasquez, Mar 24 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 the same point in A. 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 in the same row and 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 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 (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) - 4747*15^(n-3) - 166*3^(n-3) + 1534*7^(n-3) + 4.

A342796 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 the same vertex in the 6-point set.

Original entry on oeis.org

709682, 77784248, 6126191066, 427218509360, 28245026082242, 1821452259070568, 116065734824421866, 7353059854962677600, 464513906582191544402, 29303821259651224580888, 1847364138146506201033466, 116421875056692663153073040

Offset: 4

Roman I. Vasquez, Mar 24 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 the same point in A. 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 in the same row and 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 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 (120,-4593,69688,-428787,978768,-615195).

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) = 29791*63^(n-3) - 46666*31^(n-3) + 20305*15^(n-3) - 3700*7^(n-3) + 275*3^(n-3) - 5.

A342850 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 and none of the removed edges are incident to the same vertex in the other set.

Original entry on oeis.org

162, 1242, 9018, 64098, 451602, 3169962, 22215978, 155590578, 1089370242, 7626300282, 53386227738, 373709971458, 2615988932082, 18311979920202, 128184031628298, 897288737958738, 6281022715393122, 43967163656797722, 307770159544721658

Offset: 4

Roman I. Vasquez, Mar 24 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 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} 3 X n matrices (with n at least 4) 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(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 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 (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) = 27*7^(n-3) - 3^(n-1).

G.f.: 54*x^4*(3 - 7*x)/(1 - 10*x + 21*x^2). - Stefano Spezia, Mar 25 2021

A343372 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 removed edges are incident to the same vertex in the 3-point set and exactly two removed edges are incident to the same vertex in the other set.

Original entry on oeis.org

112, 922, 6880, 49450, 350032, 2461882, 17268160, 120982090, 847189552, 5931271642, 41521735840, 290660653930, 2034650086672, 14242627134202, 99698619521920, 697891025400970, 4885239244049392

Offset: 4

Steven Schlicker, Apr 12 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 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} 3 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(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 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 (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.

Drop[CoefficientList[Series[2 x^4*(56 - 155 x + 105 x^2)/(1 - 11 x + 31 x^2 - 21 x^3), {x, 0, 20}], x], 4] (* Michael De Vlieger, Apr 13 2021 *)
LinearRecurrence[{11,-31,21},{112,922,6880},20] (* Harvey P. Dale, Apr 06 2025 *)

a(n) = 3*7^(n-2) - 4*3^(n-2) + 1.

G.f.: 2*x^4*(56 - 155*x + 105*x^2)/(1 - 11*x + 31*x^2 - 21*x^3). - Stefano Spezia, Apr 13 2021

A343374 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 exactly two removed edges are incident to the same vertex in the 5-point set and exactly two removed edges are incident to the same vertex in the other set.

Original entry on oeis.org

58984, 2445394, 86336272, 2843754442, 90733504504, 2851869796354, 88998264600352, 2767824089452282, 85935878802252424, 2666013369738472114, 82676439390965238832, 2563420051241406849322, 79472778433612932113944, 2463757486872117920024674, 76378002443759735050203712

Offset: 4

Steven Schlicker, Apr 12 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 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} 5 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(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 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 (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) = 105*31^(n-2) - 217*15^(n-2) + 148*7^(n-2) - 13*3^(n-1) + 3.

G.f.: 2*x^4*(29492 - 458347*x + 3025391*x^2 - 7090641*x^3 + 4501665*x^4)/((1 - x)*(1 - 3*x)*(1 - 7*x)*(1 - 15*x)*(1 - 31*x)). - Stefano Spezia, Sep 01 2025

Typo in a(14) corrected by Georg Fischer, Dec 08 2021

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A342580 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 the same vertex in the 5-point set.

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A342796 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 the same vertex in the 6-point set.

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