A198808 -id:A198808 - cp's OEIS frontend

A198800 Number of closed paths of length n whose steps are 20th roots of unity, U_20(n).

1, 0, 20, 0, 1140, 480, 102800, 151200, 12310900, 38707200, 1812247920, 9574488000, 313983978000, 2391608419200, 62051403928800, 611744666332800, 13627749414064500, 160896284989440000, 3253345101771050000, 43527416858084016000, 829176006298475046640

Offset: 0

Simon Plouffe, Oct 30 2011

nonn

U_20(n) (comment in article) : For each m >= 1, the sequence (U_m(N)), N >= 0 is P-recursive but is not algebraic when m > 2.

Andrew Howroyd, Table of n, a(n) for n = 0..200
Gilbert Labelle and Annie Lacasse, Closed paths whose steps are roots of unity, in FPSAC 2011, Reykjavik, Iceland DMTCS proc. AO, 2011, 599-610.

Cf. A070190, A198808.

seq(n)={Vec(serlaplace(sum(k=0, n, if(k,2,1)*(x^k*besseli(k, 2*x + O(x^(n-k+1)))/k!)^5)^2))} \\ Andrew Howroyd, Nov 01 2018

E.g.f.: g(x)^2 where g(x) is the e.g.f. of A070190. - Andrew Howroyd, Nov 01 2018

a(n) ~ 2^(2*n) * 5^(n+3) / (Pi^4 * n^4). - Vaclav Kotesovec, Apr 30 2024

A321417 Number of n element multisets of the 12th roots of unity with zero sum.

Original entry on oeis.org

1, 0, 6, 4, 21, 24, 64, 84, 174, 236, 420, 576, 926, 1260, 1896, 2540, 3639, 4800, 6618, 8592, 11499, 14700, 19200, 24204, 30972, 38544, 48480, 59620, 73884, 89892, 109960, 132480, 160221, 191308, 229038, 271248, 321809, 378264, 445128, 519608, 606954, 704016

Offset: 0

Andrew Howroyd, Nov 09 2018

Equivalently, the number of closed convex paths of length n whose steps are the 12th roots of unity up to translation. For even n, there will be 6 paths of zero area consisting of n/2 steps in one direction followed by n/2 steps in the opposite direction.

Andrew Howroyd, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (2,3,-6,-6, 6,13,-2,-18, -2,13,6,-6, -6,3,2,-1)

Column k=6 of A321414.

Cf. A053090, A198808.

Vec(((2/(1 - x^3) - 1)/(1 - x^2)^3)^2 + O(x^40))

G.f.: ((2/(1 - x^3) - 1)/(1 - x^2)^3)^2.

G.f.: (1 - x + x^2)^2/((1 + x + x^2)^2*(1 - x)^8*(1 + x)^4).

A198802 Number of closed paths of length n whose steps are 18th roots of unity, U_18(n).

Original entry on oeis.org

1, 0, 18, 36, 918, 5400, 82800, 801360, 10907190, 132053040, 1802041668, 24199809480, 340640607384, 4834708246368, 70229958125184, 1032223723667136, 15391538570569590, 231935110984687968, 3531542904056225916, 54244559313713885688, 839979883121036697468

Offset: 0

Simon Plouffe, Oct 30 2011

nonn

U_18(n), comment in article: For each m >= 1, the sequence (U_m(N)), N >= 0 is P-recursive but is not algebraic when m > 2.

Andrew Howroyd, Table of n, a(n) for n = 0..200
Gilbert Labelle and Annie Lacasse, Closed paths whose steps are roots of unity, in FPSAC 2011, Reykjavik, Iceland DMTCS proc. AO, 2011, 599-610.

Cf. A002898, A198808.

seq(n)={Vec(serlaplace(sum(k=0, n, if(k,2,1)*(x^k*besseli(k, 2*x + O(x^(n-k+1)))/k!)^3)^3))} \\ Andrew Howroyd, Nov 01 2018

E.g.f.: g(x)^3 where g(x) is the e.g.f. of A002898.

a(n) ~ 2^(n-3) * 3^(2*n + 9/2) / (Pi^3 * n^3). - Vaclav Kotesovec, Apr 30 2024

A198805 Number of closed paths of length n whose steps are 15th roots of unity, U_15(n).

Original entry on oeis.org

1, 0, 0, 30, 0, 360, 7650, 0, 302400, 4544400, 11226600, 324324000, 4310633250, 24324300000, 437404968000, 5634178329780, 45972927000000, 697866761592000, 8962716395833200, 88725951057744000, 1258898645656852200

Offset: 0

Simon Plouffe, Oct 30 2011

nonn

U_15(n), (comment in article): For each m >= 1, the sequence (U_m(N)), N >= 0 is P-recursive but is not algebraic when m > 2.

Gilbert Labelle and Annie Lacasse, Closed paths whose steps are roots of unity, in FPSAC 2011, Reykjavik, Iceland DMTCS proc. AO, 2011, 599-610.

Cf. A198808, A321418.

cp's OEIS Frontend

A198800 Number of closed paths of length n whose steps are 20th roots of unity, U_20(n).

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A321417 Number of n element multisets of the 12th roots of unity with zero sum.

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A198802 Number of closed paths of length n whose steps are 18th roots of unity, U_18(n).

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A198805 Number of closed paths of length n whose steps are 15th roots of unity, U_15(n).

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