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.

User: Ivaylo Kortezov

Ivaylo Kortezov's wiki page.

Ivaylo Kortezov has authored 17 sequences. Here are the ten most recent ones:

A369810 Number of ways to color n+1 identical balls using n distinct colors (each color is used) and place them in n numbered cells so that each cell contains at least one ball.

Original entry on oeis.org

1, 8, 63, 528, 4800, 47520, 511560, 5967360, 75116160, 1016064000, 14709340800, 227046758400, 3723758438400, 64686292070400, 1186714488960000, 22931377717248000, 465594843377664000, 9910874496466944000, 220725034691825664000, 5133423237252710400000
Offset: 1

Views

Author

Ivaylo Kortezov, Feb 02 2024

Keywords

Examples

			For n=3 one of the colors c (3 choices) is used twice and one of the cells k (3 choices) gets two balls. If the cell k does not contain a c-colored ball, then all other cells do (1 variant). If the cell k contains a c-colored ball, after its removal there are 3!=6 variants for placing the remaining 3 different balls in the 3 cells. In total there are 3*3*(1+6)=63 variants.

Crossrefs

Cf. A006595, A284816.

Programs

Formula

a(n) = n!*n*(n^2+n+2)/4.
a(n) = n*A284816(n).
a(n) = n^2*A006595(n-1).
E.g.f.: x*(2 + x^2)/(2*(1 - x)^4). - Stefano Spezia, Feb 05 2024

A363964 Number of unordered pairs of non-intersecting non-self-intersecting paths, singletons included, with nodes that cover all vertices of a convex labeled n-gon.

Original entry on oeis.org

3, 14, 55, 195, 644, 2016, 6048, 17520, 49280, 135168, 362752, 955136, 2472960, 6307840, 15876096, 39481344, 97124352, 236584960, 571146240, 1367539712, 3249799168, 7669284864, 17983078400, 41916825600, 97165246464, 224076496896, 514272002048, 1174992322560
Offset: 3

Views

Author

Ivaylo Kortezov, Jun 30 2023

Keywords

Comments

For each such path there is a sequence of distinct vertices of the n-gon, each (except the last one) connected by a segment with the next vertex in the sequence; the segments have no common internal points. The path itself is the union of the set of these segments and is thus direction-independent: reversing the order of the vertices leads to the same path. If the sequence of vertices has length 1 then there are no segments; we call such a path a singleton.

Examples

			a(4)=14 since if one of the paths is a singleton (4 choices), then there are A001792(3)=3 choices for the other path, and otherwise for the two paths there are A308914(4)=2 choices, so a(4)=4*3+2=14.

Crossrefs

Cf. A001792, A308914, A332426, A360716, A360717.

Formula

a(n) = n*(n-1)*(n^2+n+36)*2^(n-8)/3.

A362786 Number of unordered triples of disjoint self-avoiding paths with nodes that cover all vertices of a convex n-gon.

Original entry on oeis.org

0, 0, 0, 5, 63, 476, 2772, 13680, 60060, 241472, 906048, 3214848, 10890880, 35481600, 111794176, 342171648, 1021031424, 2979102720, 8520171520, 23934468096, 66156625920, 180198047744, 484304486400, 1285790105600, 3375480176640, 8769899593728, 22567515586560, 57557594931200
Offset: 3

Views

Author

Ivaylo Kortezov, May 04 2023

Keywords

Examples

			For n=7 we have one 3-node path and two 2-node paths. Call two paths adjacent if we can choose one node from each path so that the two nodes are adjacent vertices of the n-gon. Then either each pair of paths is adjacent, or the two 2-node paths are not adjacent, or a 2-node path is not adjacent to the 3-node path. In each of these three cases there are 7 choices for the set of nodes for the 3-node path and 3 ways to connect them, and then the 2-node paths are uniquely determined. Thus a(7) = 3*7*3 = 63.

Crossrefs

The number of unordered pairs of disjoint self-avoiding paths with nodes that cover all vertices of a convex n-gon is A308914(n). The number of unordered triples of (not necessarily disjoint) self-avoiding paths with nodes that cover all vertices of a convex n-gon is A359404(n).

Formula

a(n) = 2^(n-12)*n*(n-1)*(n-2)*(n-4)*(n-5)*(n+2)*(n+9)/90 for n > 3; 0 for n=3.

A361284 Number of unordered triples of self-avoiding paths whose sets of nodes are disjoint subsets of a set of n points on a circle; one-node paths are not allowed.

Original entry on oeis.org

0, 0, 0, 0, 0, 15, 420, 7140, 95760, 1116990, 11891880, 118776900, 1132182480, 10415938533, 93207174060, 815777235000, 7011723045600, 59364660734172, 496238466573648, 4102968354298200, 33602671702168800, 272909132004479355, 2200084921469527092, 17618774018675345340, 140252152286127750000
Offset: 1

Views

Author

Ivaylo Kortezov, Mar 07 2023

Keywords

Comments

Although each path is self-avoiding, the different paths are allowed to intersect.

Examples

			a(7) = A359404(7) + 7*A359404(6) = 315 + 7*15 = 420 since either all the 7 points are used or one is not.

Links

Crossrefs

If there is only one path, we get A261064. If there is are two paths, we get A360716. If all n points need to be used, we get A359404.

Programs

  • PARI
    a(n) = {(n*(n-1)*(n-2)/384) * (7^(n-3) - 3*5^(n-3) + 3^(n-2) - 1)} \\ Andrew Howroyd, Mar 07 2023

Formula

a(n) = (n*(n-1)*(n-2)/384)*(7^(n-3) - 3*5^(n-3) + 3^(n-2) - 1).
E.g.f.: x^3*exp(x)*(exp(2*x) - 1)^3/384. - Andrew Howroyd, Mar 07 2023

A361285 Number of unordered triples of self-avoiding paths whose sets of nodes are disjoint subsets of a set of n points on a circle; one-node paths are allowed.

Original entry on oeis.org

0, 0, 1, 10, 85, 695, 5600, 45080, 364854, 2973270, 24382875, 200967250, 1662197251, 13772638789, 114126098450, 944285871200, 7791140945180, 64038240953196, 523977421054245, 4266101869823850, 34554155058753505, 278417272387723315, 2231755184899383220, 17799741659621513240
Offset: 1

Views

Author

Ivaylo Kortezov, Mar 07 2023

Keywords

Comments

Although each path is self-avoiding, the different paths are allowed to intersect.

Examples

			a(4) = A360021(4) + 4*A360021(3) = 6 + 4 = 10 since either all the 4 points are used or one is not.

Links

Crossrefs

If there is only one path, we get A360715. If there is are two paths, we get A360717. If all n points need to be used, we get A360021.

Programs

  • PARI
    a(n) = {(n*(n-1)*(n-2)/384) * (7^(n-3) + 9*5^(n-3) + 3^n + 27)} \\ Andrew Howroyd, Mar 07 2023

Formula

a(n) = (n*(n-1)*(n-2)/384)*(7^(n-3) + 9*5^(n-3) + 3^n + 27).
E.g.f.: x^3*exp(x)*(exp(2*x) + 3)^3/384. - Andrew Howroyd, Mar 07 2023

A360717 Number of unordered pairs of self-avoiding paths whose sets of nodes are disjoint subsets of a set of n points on a circle; one-node paths are allowed.

Original entry on oeis.org

0, 1, 6, 33, 185, 1050, 6027, 35014, 205326, 1209375, 7119860, 41744703, 243218703, 1406685280, 8073640785, 45991600860, 260131208396, 1461591509805, 8162196518322, 45327133739245, 250431036147285, 1377169337010390, 7540979990097191, 41130452834689218, 223528009015333050, 1210753768099880875, 6537995998163877312
Offset: 1

Views

Author

Ivaylo Kortezov, Feb 18 2023

Keywords

Comments

Although each path is self-avoiding, the different paths are allowed to intersect.

Examples

			a(4) = A359405(4) + 4*A359405(3) + 4*3/2 = 15 + 12 + 6 = 33 with the three summands corresponding to the cases of 4, 3 and 2 used points.

Links

Crossrefs

If there is only one path, we get A360715. If one-node paths are not allowed, we get A360716.

Formula

a(n) = n*(n-1)*2^(-5)*(5^(n-2) + 6*3^(n-2) + 9).
E.g.f.: exp(x)*((x*exp(2*x) + 3*x)/4)^2/2. - Andrew Howroyd, Feb 19 2023

A360715 Number of self-avoiding paths with nodes chosen among n given points on a circle; one-node paths are allowed.

Original entry on oeis.org

1, 3, 9, 30, 105, 369, 1281, 4380, 14769, 49215, 162393, 531450, 1727193, 5580141, 17936145, 57395640, 182948577, 581130747, 1840247337, 5811307350, 18305618121, 57531942633, 180441092769, 564859072980, 1765184603025, 5507375961399, 17157594341241, 53379182394930, 165856745298489, 514727830236645
Offset: 1

Views

Author

Ivaylo Kortezov, Feb 18 2023

Keywords

Examples

			a(4) = A001792(2) + 4*A001792(1) + 6 + 4 = 8 + 4*3 + 6 + 4 = 30 with the four summands corresponding to paths with 4, 3, 2 and 1 nodes, respectively.

Links

Crossrefs

If one-node paths are not allowed, one gets A261064. Cf. A001792 if all n points need to be used.

Formula

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

A360716 Number of unordered pairs of self-avoiding paths whose sets of nodes are disjoint subsets of a set of n points on a circle; one-node paths are not allowed.

Original entry on oeis.org

0, 0, 0, 3, 45, 435, 3465, 24794, 165942, 1061730, 6578550, 39796053, 236309931, 1382504669, 7989938775, 45704622660, 259155482652, 1458298435572, 8151155034300, 45290328792695, 250308998693145, 1376766613411959, 7539656755416885, 41126122248463038, 223513887538508850, 1210707873300202550, 6537847299012919890
Offset: 1

Views

Author

Ivaylo Kortezov, Feb 18 2023

Keywords

Comments

Although each path is self-avoiding, the different paths are allowed to intersect.

Examples

			a(5)=30+15=45: the first summand corresponds to the case when one of the paths has three nodes (5*4*3/2=30 variants; division by 2 is due to directional independence) and the second to the case when both paths have two nodes (5!/(2!2!2!)=15 variants).

Links

Crossrefs

If there is only one path, we get A261064. If all n points need to be used, we get A332426.

Formula

a(n) = n*(n-1)*2^(-5)*(5^(n-2) - 2*3^(n-2) + 1).
From Andrew Howroyd, Feb 19 2023: (Start)
Binomial transform of A332426.
a(n) = 27*a(n-1) - 312*a(n-2) + 2016*a(n-3) - 7986*a(n-4) + 19998*a(n-5) - 31472*a(n-6) + 29880*a(n-7) - 15525*a(n-8) + 3375*a(n-9) for n > 9.
G.f.: x^4*(3 - 36*x + 156*x^2 - 288*x^3 + 197*x^4)/((1 - x)*(1 - 3*x)*(1 - 5*x))^3.
E.g.f.: exp(x)*(exp(2*x) - 1)^2*x^2/32.
(End)

A360275 Number of unordered quadruples of self-avoiding paths with nodes that cover all vertices of a convex n-gon.

Original entry on oeis.org

0, 0, 0, 0, 0, 105, 3780, 81900, 1386000, 20207880, 266666400, 3277354080, 38198160000, 427365818880, 4629059635200, 48842864179200, 504335346278400, 5114054709319680, 51064119467827200, 503151159589478400, 4900668252598272000, 47248486914198011904, 451429610841538560000
Offset: 3

Views

Author

Ivaylo Kortezov, Feb 01 2023

Keywords

Comments

The paths considered here cover at least 2 vertices. Although each path is self-avoiding, the different paths are allowed to intersect.

Examples

			a(9) = 9!*3/(2!2!2!3!3!) = 3780 since we have to split the 9 vertices into three pairs and one triple, the order of the three pairs is irrelevant, and there are 3 ways of connecting the triple.

Links

Crossrefs

Cf. A001792, A332426 (unordered pairs of paths), A359404 (unordered triples of paths).

Formula

a(n) = (1/3)*n*(n-1)*(n-2)*(n-3)*2^(n-15)*(4^(n-4) - 4*3^(n-4) + 6*2^(n-4) - 4) for n != 4.

A360276 Number of unordered quadruples of self-avoiding paths with nodes that cover all vertices of a convex n-gon; one-node paths are allowed.

Original entry on oeis.org

0, 0, 10, 105, 1015, 9625, 90972, 861420, 8191920, 78309000, 752317280, 7257522272, 70223986560, 680703296000, 6601793730560, 63984047339520, 619018056228864, 5972223901440000, 57415027394027520, 549677356175073280, 5238367168966328320, 49678823782558924800, 468783944069762252800
Offset: 3

Views

Author

Ivaylo Kortezov, Feb 01 2023

Keywords

Comments

Although each path is self-avoiding, the different paths are allowed to intersect.

Examples

			a(6) = 6!/(2!2!2!2!)+6!*3/(3!3!) = 45+60 = 105; the first summand corresponds to the case of 2 two-node paths and 2 one-node paths; the second to the case of 1 three-node path and 3 one-node paths.

Links

Crossrefs

Cf. A001792, A359405 (unordered pairs of paths), A360021 (unordered triples of paths).

Formula

a(n) = (1/3)*n*(n-1)*(n-2)*(n-3)*2^(n-15)*(4^(n-4) + 12*3^(n-4) + 54*2^(n-4) + 108) for n != 4.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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