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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: Jacob Landgraf

Jacob Landgraf's wiki page.

Jacob Landgraf has authored 2 sequences.

A260746 Number of prime juggling patterns of period n using 4 balls.

Original entry on oeis.org

1, 4, 19, 83, 391, 1663, 7739, 33812, 153575, 677901, 3075879, 13586581, 61458267, 272367077, 1228519987, 5456053443, 24547516939, 109153816505, 490067180301, 2180061001275, 9772927018285, 43467641569472
Offset: 1

Views

Author

Jacob Landgraf, Jul 30 2015

Keywords

Comments

A juggling pattern is prime if the closed walk corresponding to the pattern in the juggling state graph is a cycle.

Examples

			In siteswap notation, the prime juggling pattern(s) of length one is 4; of length two are 53, 62, 71 and 80; of length three are (11)01, (12)00, 660, 750, (10)11, (10)20, 390, 831, 822, 471, 561, 741, 723, 633, 642, 552, 912, 930 and 480.

Links

Crossrefs

Cf. A260744, A260745, A260752.

Extensions

a(14) from Roman Berens, Mar 20 2021
a(15)-a(22) from Jack Boyce, May 31 2024

A260584 Number of ways to place 4n rooks on n X n board, 4 rooks in each row and each column, multiple rooks in a cell allowed, and exactly 4 rooks below the main diagonal.

Original entry on oeis.org

1, 42, 1152, 22785, 358784, 4848569, 59160195, 674020718, 7332379979, 77311947872, 798116114567, 8122264310217, 81865063934240, 819786478839348, 8173571362926773, 81256681626746819, 806240597786756436, 7989356540290573170
Offset: 2

Views

Author

Jacob Landgraf, Jul 29 2015

Keywords

Comments

a(n) is the number of minimal multiplex juggling patterns of period n using exactly 4 balls when we can catch/throw up to 4 balls at a time. (Minimal in the sense that the throws are between 0 and n-1.)

Links

Crossrefs

Cf. A260575, A260582, A260583, A260585, A260727.

Programs

  • Mathematica
    CoefficientList[Series[-(25340000 x^23 -218339000 x^22 + 967516125 x^21 - 3006320955 x^20 + 7236534214 x^19 - 13729556248 x^18 + 20397063058 x^17 - 23597394968 x^16 + 21251854412 x^15 - 14962982713 x^14 + 8335966059 x^13 - 3793119227 x^12 + 1513380019 x^11 - 584800410 x^10 + 226357446 x^9 - 80585779 x^8 + 23590993 x^7 - 5268629 x^6 + 855872 x^5 - 97502 x^4 + 7700 x^3 - 464 x^2 + 26 x^1 - 1)/(196000000 x^26 - 2903600000 x^25 + 20460490000 x^24 - 91266464000 x^23 + 289327787000 x^22 - 693785336400 x^21 + 1307696973825 x^20 - 1987649503130 x^19 + 2479934403745 x^18 - 2572088215962 x^17 + 2237510543313 x^16 - 1642726164623 x^15 + 1021902480875 x^14 - 539757845397 x^13 + 242151721153 x^12 - 92151943921 x^11 + 29657096575 x^10 - 8031745172 x^9 + 1817290072 x^8 - 340120209 x^7 + 51938261 x^6 - 6350073 x^5 + 605172 x^4 - 43205 x^3 + 2168 x^2 -68 x + 1), {x, 0, 20}], x] (* Vincenzo Librandi, Aug 06 2015 *)

Formula

G.f.: -(25340000*x^25 - 218339000*x^24 + 967516125*x^23 - 3006320955*x^22 + 7236534214*x^21 - 13729556248*x^20 + 20397063058*x^19 - 23597394968*x^18 + 21251854412*x^17 - 14962982713*x^16 + 8335966059*x^15 - 3793119227*x^14 + 1513380019*x^13 - 584800410*x^12 + 226357446*x^11 - 80585779*x^10 + 23590993*x^9 - 5268629*x^8 + 855872*x^7 - 97502*x^6 + 7700*x^5 - 464*x^4 + 26*x^3 - x^2)/(196000000*x^26 - 2903600000*x^25 + 20460490000*x^24 - 91266464000*x^23 + 289327787000*x^22 - 693785336400*x^21 + 1307696973825*x^20 - 1987649503130*x^19 + 2479934403745*x^18 - 2572088215962*x^17 + 2237510543313*x^16 - 1642726164623*x^15 + 1021902480875*x^14 - 539757845397*x^13 + 242151721153*x^12 - 92151943921*x^11 + 29657096575*x^10 - 8031745172*x^9 + 1817290072*x^8 - 340120209*x^7 + 51938261*x^6 - 6350073*x^5 + 605172*x^4 - 43205*x^3 + 2168*x^2 - 68*x + 1).

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